MATH/STAT. 


AN  ELEMENTARY  TREATISE 


ON 


DIFFERENTIAL   EQUATIONS 


BY 


ABRAHAM    COHEN,   PH.D. 

ASSOCIATE   PROFESSOR   OF  MATHEMATICS 
JOHNS  HOPKINS   UNIVERSITY 


D.    C.    HEATH    &   CO.,    PUBLISHERS 

BOSTON        NEW  YORK        CHICAGO 


BY    THE    SAME    AUTHOR 


An  Introduction  to  the  Lie  Theory 
of  One-Parameter  Groups 


vii  +  248  pages         Half  Leather 


D.    C.    HEATH    &    CO.,    PUBLISHERS 


COPYRIGHT,  1906, 
BY  D.  C.  HEATH  &  Co. 

3Kl 


Printed  in  U.  S.  A. 


QA17! 
CUI 


PREFACE 

THE  following  pages  are  the  result  of  a  course  in  Differential 
Equations  which  the  author  has  given  for  some  years  to  classes 
comprising  students  intending  to  pursue  the  study  of  Engineering 
or  some  other  Physical  Science,  as  well  as  those  expecting  to  con 
tinue  the  study  of  Pure  Mathematics  or  Mathematical  Physics. 

The  primary  object  of  this  book  is  to  make  the  student  familiar 
with  the  principles  and  devices  that  will  enable  him  to  integrate 
most  of  the  equations  he  is  apt  to  come  across.  As  much  of  the 
theory  is  given  as  is  likely  to  be  comprehensible  to  the  student  who 
has  had  a  year's  course  in  the  Differential  and  Integral  Calculus, 
and  yet  is  sufficient  to  form  a  harmonizing  setting  for  the  numerous 
and  otherwise  apparently  miscellaneous  classes  of  equations,  and 
the  disconnected  methods  for  solving  them.  It  is  intended  to  have 
the  work  sufficiently  broad  to  make  it  a  handy  book  of  reference, 
without  affecting  its  utility  as  a  text-book.  A  number  of  footnotes 
and  remarks  have  been  put  in,  which,  without  breaking  the  continu 
ity  of  the  practical  side  of  the  subject,  must  prove  of  interest  and 
value.  Numerous  historical  and  bibliographical  references  are 
also  made. 

A  course  that  is  limited  in  point  of  time  and  aims  only  at  acquir 
ing  skill  in  integrating  most  of  the  equations  that  are  apt  to  arise 
could  dispense  with  §§  12,  15,  17,  22,  28  (part  in  small  type),  33, 
34,  38-40,  46-48,  66-69  (except  examples),  70,  71,  73,  75,  78,  80, 
81.  Many  of  these  sections  should  properly  come  in  a  well-bal 
anced  course.  The  needs  of  the  class,  and  the  time  at  its  disposal, 
must  decide  which  of  them,  if  any,  should  be  omitted. 

The  author  has  had  in  mind  continually  the  necessity  of  sys 
tematizing  the  various  classes  of  equations  that  can  be  solved  by 
elementary  means,  and  of  minimizing  the  number  of  methods  by 
which  they  can  be  solved.  To  enable  the  student  to  get  a  better 

ill 


IV  PREFACE 

general  view  of  the  subject,  the  summaries  at  the  ends  of  the 
various  chapters  and  the  final  general  summary  must  prove  of 
great  value. 

Numerous  applications  to  problems  in  Geometry  and  the  Phys 
ical  Sciences  have  been  introduced,  both  in  the  body  of  the  text 
and  in  the  form  of  exercises  for  the  student. 

Although  a  large  number  of  the  problems  have  been  published 
before,  many  are  new,  and  all  have  been  chosen  to  bring  out  the 
various  methods  of  the  differential  equations,  and  of  the  integral 
calculus  as  well.  Many  of  the  examples  worked  out  in  the  text 
were  chosen  to  recall  some  of  the  more  important  methods  of  the 
latter;  for  while  the  use  of  tables  of  integrals  is  recommended, 
the  student  should  not  feel  absolutely  dependent  upon  them. 
Most  of  the  solutions  have  a  simple  form  or  an  interesting  inter 
pretation.  Great  care  has  been  taken  to  avoid  typographical 
errors.  The  author  shall  be  very  glad  to  learn  of  any  that  still 
exist. 

The  method  of  undetermined  coefficients  for  finding  the  particu 
lar  integral  in  the  case  of  linear  equations  with  constant  coefficients 
is  believed  to  be  presented  here  for  the  first  time  in  its  complete 
form. 

The  subject  of  Partial  Differential  Equations  is  so  vast  that  it  was 
decided  to  present  only  a  few  topics,  which,  in  all  probability,  will 
suffice  for  the  needs  of  the  students  for  whom  this  book  is  intended. 

It  was  only  after  considerable  thought  that  the  author  refrained 
from  adding  a  chapter  on  the  Lie  Theory.  It  is  hoped  to  present 
that  important  branch  of  the  subject  in  a  separate  volume. 

In  conclusion  the  author  takes  great  pleasure  in  expressing  his 
appreciation  of  the  valuable  suggestions  made  by  Professor  F.  S. 
Woods  of  the  Massachusetts  Institute  of  Technology,  as  well  as 
of  those  by  Professor  L.  G.  Weld  of  the  University  of  Iowa  and 
Professor  E.  J.  Townsend  of  the  University  of  Illinois. 

ABRAHAM   COHEN. 

JOHNS  HOPKINS  UNIVERSITY,  BALTIMORE,  MARYLAND, 
October,  1906. 


CONTENTS 

CHAPTER   I 
DIFFERENTIAL  EQUATIONS  AND  THEIR  SOLUTIONS 

SECTION  PAGE 

1.  Differential  Equation.    Ordinary  and  Partial.     Order,  Degree       .         .  I 

2.  Solution  of  an  Equation          .........  2 

3.  Derivation  of  a  Differential  Equation  from  its  Primitive         ...  3 

4.  General,  Particular  Solution 5 

CHAPTER   II 
DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER  AND  FIRST  DEGREE 

5.  Exact  Differential  Equation.     Integrating  Factor           ....  7 

6.  General  Plan  of  Solution 9 

7.  Condition  that  Equation  be  Exact 9 

8.  Exact  Differential  Equations II 

9.  Variables  Separated  or  Separable 13 

10.  Homogeneous  Equations 14 

11.  Equations  in  which  M  and  N are  Linear  but  not  Homogeneous    .         .  16 

12.  Equations  of  the  Form  yf\(xy)dx  -f  xfo(xy} dy  —  o        .         .         .  17 

13.  Linear  Equations  of  the  First  Order 18 

14.  Equations  Reducible  to  Linear  Equations 20 

15.  Equations  of  the  Form  xry*(niy  dx  +  nx  dy)-\-x?y*  (jty  dx -f  vx  dy)  =  o  .  22 

1 6.  Integrating  Factors  by  Inspection 23 

17.  Other  Forms  for  which  Integrating  Factors  can  be  Found     ...  24 

1 8.  Transformation  of  Variables 26 

19.  Summary 28 

CHAPTER   III 

APPLICATIONS 

20.  Differential  Equation  of  a  Family  of  Curves 31 

21.  Geometrical  Problems  involving  the  Solution  of  Differential  Equations  34 


VI  CONTENTS 

SECTION  PAGE 

22.  Orthogonal  Trajectories 38 

23.  Physical  Problems  giving  Rise  to  Differential  Equations        ...  43 

CHAPTER   IV 

DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER  AND  HIGHER 
DEGREE  THAN  THE  FIRST 

24.  Equations  Solvable  for/ 49 

25.  Equations  Solvable  for  y         .........  52 

26.  Equations  Solvable  for  x        .........  cc 

27.  Clairaut's  Equation         ..........  56 

28.  Summary 58 

CHAPTER  V 

SINGULAR  SOLUTIONS 

29.  Envelopes 61 

30.  Singular  Solutions 63 

31.  Discriminant           ...........  64 

32.  Singular  Solution  Obtained  directly  from  the  Differential  Equation        .  66 

33.  Extraneous  Loci 69 

34.  Summary 75 

CHAPTER   VI 

TOTAL  DIFFERENTIAL  EQUATIONS 

35.  Total  Differential  Equations 76 

36.  Method  of  Solution 80 

37.  Homogeneous  Equations 81 

38.  Equations  involving  more  than  Three  Variables 83 

39.  Equations  which  do  not  satisfy  the  Condition  for  Integrability       .         .  84 

40.  Geometrical  Interpretation     .........  86 

41.  Summary        ............  87 

CHAPTER   VII 
LINEAR  DIFFERENTIAL  EQUATIONS  WITH  CONSTANT  COEFFICIENTS 

42.  General  Linear  Differential  Equation 89 

43.  Linear  Differential  Equations  with  Constant  Coefficients.     Complemen 

tary  Function 91 


CONTENTS  vii 

SECTION  PAGE 

44.  Roots  of  Auxiliary  Equation  Repeated 93 

45.  Roots  of  the  Auxiliary  Equation  Complex 94 

46.  Properties  of  the  Symbolic  Operator  (Z>  —  a) 96 

47.  Particular  Integral 97 

48.  Another  Method  of  rinding  the  Particular  Integral         ....  101 

49.  Variation  of  Parameters 103 

50.  Method  of  Undetermined  Coefficients 107 

51.  Cauchy's  Linear  Equation 113 

52.  Summary 115 

CHAPTER   VIII 
LINEAR  DIFFERENTIAL  EQUATIONS  OF  THE  SECOND  ORDER 

53.  Change  of  Dependent  Variable 123 

54.  Change  of  Independent  Variable 127 

55.  Summary 129 

CHAPTER   IX 

MISCELLANEOUS  METHODS  FOR  SOLVING  EQUATIONS  OF  HIGHER 
ORDER  THAN  THE  FIRST 

56.  General  Plan  of  Solution 131 

57.  Dependent  Variable  Absent 131 

58.  Independent  Variable  Absent 134 

59.  Linear  Equations  with  Particular  Integral  Known          .         .         .         .  135 

60.  Exact  Equation.     Integrating  Factor 137 

61.  Transformation  of  Variables 143 

62.  Summary 144 

CHAPTER  X 

SYSTEMS  OF  SIMULTANEOUS  EQUATIONS 

63.  General  Method  of  Solution 149 

64.  Systems  of  Linear  Equations  with  Constant  Coefficients          .         .         .  150 

65.  Systems  of  Equations  of  the  First  Order 153 

66.  Geometrical  Interpretation 157 

67.  Systems  of  Total  Differential  Equations 159 

68.  Differential  Equations  of  Higher  Order  than  the  First   Reducible   to 

Systems  of  Equations  of  the  First  Order 1 60 

69.  Summary 161 


Vlll  CONTENTS 


CHAPTER  XI 

INTEGRATION  IN  SERIES 

SECTION  PAGB 

70.  The  Existence  Theorem 164 

71.  Singular  Solutions 168 

72.  Integration,  in  Series,  of  an  Equation  of  the  First  Order        .         .         .  169 

73.  Riccati's  Equation 173 

74.  Integration,  in  Series,  of  Equations  of  Higher  Order  than  the  First        .  177 

75.  Gauss's  Equation.     Hypergeometric  Series 192 

CHAPTER   XII 

PARTIAL  DIFFERENTIAL  EQUATIONS 

76.  Primitives  involving  Arbitrary  Constants 196 

77.  Primitives  involving  Arbitrary  Functions 199 

78.  Solution  of  a  Partial  Differential  Equation 202 

CHAPTER   XIII 

PARTIAL  DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER 

79.  Linear  Partial  Differential  Equations  of  the  Plrst  Order.     Method  of 

Lagrange 205 

So.    Integrating  Factors  of  the  Ordinary  Differential  Equation  M  dx+Ndy 

—  o   . 209 

81.  Non-linear   Partial  Differential  Equations  of  First  Order.     Complete, 

General,  Singular  Solutions 211 

82.  Method  of  Lagrange  and  Charpit 215 

83.  Special  Methods 221 

84.  Summary 227 

CHAPTER   XIV 

PARTIAL  DIFFERENTIAL  EQUATIONS  OF  HIGHER  ORDER  THAN  THE 

FIRST 

85.  Partial  Differential  Equations  of  the  Second  Order,  Linear  in  the  Second 

Derivatives.     Monge's  Method 230 

86.  Special  Method 236 

87.  General  Linear  Partial  Differential  Equations 238 

88.  Homogeneous  Linear  Equations  with  Constant  Coefficients  .         .  239 


CONTENTS  ix 

SECTION  PAGK 

89.  Roots  of  Auxiliary  Equation  Repeated 240 

90.  Roots  of  Auxiliary  Equation  Complex 242 

91.  Particular  Integral 243 

92.  Non-homogeneous  Linear  Equations  with  Constant  Coefficients     .         .  246 

93.  Equations  Reducible  to  Linear  Equations  with  Constant  Coefficients     .  249 

94.  Summary 2SI 

NOTES 

I.     Condition  that  a  Relation  exist  between  Two  Functions  of  Two  Variables  253 

II.     General  Summary 254 

ANSWERS  TO  EXAMPLES 257 

INDEX 269 

REFERENCES ' 27J 


DIFFERENTIAL    EQUATIONS 

CHAPTER    I 
DIFFERENTIAL  EQUATIONS  AND  THEIR  SOLUTIONS 

1.    Differential  Equation.     Ordinary  and  Partial.     Order,  Degree. 
•A  differential  equation  is  an   equation  involving    differentials    or 
derivatives.     Thus, 

(0 


(2)  (x-yy=a? 

(  \  dy  ,      dx 

(3)  y-x-/-  +  3-r 

dx         dy 


(4) 

dx     y  dy 

//r\  d2z  .          d2z        dz 

dx2          dxdy      dy 

(7) 


are  examples  of  differential  equations. 

Equations  in  which  there  is  a  single  independent  variable  (and 
which,  therefore,  involve  ordinary  derivatives)  are  known  as  ordinary 
differential  equations.  Equations  (i),  (2),  (3),  (4),  (7)  are  such. 

If  an  equation  involves  more  than  one  independent  variable,  so 
that  partial  derivatives  enter,  it  is  known  as  a  partial  differential 
equation,  Examples  of  such  are  equations  (5),  (6). 

I 


2  DIFFERENTIAL   EQUATIONS  §2 

By  the  order  of  an  equation  we  mean  the  order  of  the  highest 
derivative  involved.  Thus,  equations  (2),  (3),  (5),  (7),  are  of  the 
first  order;  (i),  (4),  (6),  are  of  the  second  order. 

By  the  degree  of  an  equation,  we  mean  the  degree  of  the  highest 
ordered  derivative  entering,  when  the  equation  is  rationalized  and 
cleared  of  fractions  with  regard  to  all  the  derivatives.  Thus  (i),  (5), 
(6),  (7),  are  of  the  first  degree;  (2),  (3),  (4),  are  of  the  second 
degree. 

2.  Solution  of  an  Equation.  —  By  a  solution  of  a  differential  equa 
tion  we  mean  a  relation  connecting  the  dependent  and  independent 
variables  which  satisfies  the  equation.  Thus  y  =  sin  ax  is  a  solu 

tion  of  (i),  x2  +y  =  -^  is  a  solution  of  (4),  z  =  x+y  is  a  solution 
fc 


of  (5),  x*f—  i  is  a  solution  of  (7).     [The  student,  as  an  exercise, 
should  verify  these  facts.] 

Attention  should  be  called  to  the  fact  that  a  differential  equation 
has  an  indefinite  number  of  solutions.  It  can  be  seen  readily  that 
y  =  2  sin  ax,  y  =  6  cos  ax,  y  =  A  cos  ax  +  B  sin  ax  (where  A  and  B 
are  any  constants  whatever)  all  satisfy  equation  (i).  The  student  is 
in  the  habit  of  adding  a  constant  of  integration  when  integrating  a 
function.  He  says  the  integral  of  cos  x  is  sin  x  -f  c.  Now  the  prob 
lem  of  integration  studied  in  the  Calculus  is  only  a  special  case  of 
the  general  problem  of  solving  a  differential  equation.  To  integrate 

I  cos  x  dx  is  to  find  a  function,  sin  x  -\-  c,  whose  derivative  is  cos  x. 
In  the  language  of  the  Differential  Equations,  we  should  say  that  the 
solution  of  -=£  =  cos  x  is  7=  sin  x  -f-  ^,  where  c  is  an  arbitrary  constant.* 

CISC 
A  constant  in  a  solution  will  be  said  to  be  arbitrary  if  any  value  whatever 

may  be  assigned  to  it.     Thus  y  —  sin  x  +  c  is  a  solution  of  -^  =  cos  x,  no  matter 

dx 

*  It  may  be  noted  that  in  the  special  case  occurring  in  the  Integral  Calculus  the 
arbitrary  constant  always  occurs  as  an  additive  one,  while  in  the  general  case  it  may 
enter  in  an  endless  number  of  ways. 


§3         DIFFERENTIAL  EQUATIONS  AND  THEIR   SOLUTIONS  3 

what  value  c  has.  Similarly  y  —  A  cos  ax  +  B  sin  ax  is  a  solution  of  (i)  for 
any  values  of  A  and  B.  They  are  arbitrary  constants.  On  the  other  hand,  a  is 
not  arbitrary.  While  any  value  one  pleases  may  be  assigned  to  it  in  (i),  once 
chosen,  its  value  is  fixed,  and  that  value  alone  can  enter  into  the  expression  for 
the  solution. 

Restricting  ourselves  to  ordinary  differential  equations  *  we  see  that 
a  solution  may  involve  one  or  more  arbitrary  constants.  The  ques 
tion  naturally  arises,  what  is  the  maximum  number  of  such  constants 
a  solution  may  contain  ? 

3.  Derivation  of  a  Differential  Equation  from  its  Primitive.  —  Just 
as  the  problem  of  integration  is  the  inverse  of  that  of  differentiation, 
so  the  problem  of  finding  the  solution  of  a  differential  equation  is  the 
inverse  of  that  of  finding  the  differential  equation  which  is  satisfied 
by  a  relation  among  a  set  of  variables,  which  relation  may  or  may 
not  involve  one  or  more  arbitrary  constants.  In  order  to  make  this 
problem  precise,  we  shall  say  that  we  wish  to  find  the  differential 
equation  of  lowest  order  satisfied  by  this  relation,  and  not  involving 
any  arbitrary  constants.  Thus  y  =  A  cos  x,  where  A  is  an  arbitrary 

constant,    satisfies    -^  +  y  tan  x  =  o,  —£  +  y  —  o,  — 2  —  y  tan  x  =  o, 
dx  do?  dx? 

etc.     But  we  shall  say  that  -2-  -f  y  tan  x  =  o  is  the  differential  equation 

uX 

to  which  it  gives  rise. 

Again,  y  =  A  cos  x  -f  B  sin  x,  where  A  and  B  are  arbitrary  con 
stants,  satisfies  -£  +  y  =  o,  also  — ^  +  -2-  =  o,  etc.  Here,  as  before, 

«**  ax?     ax 

d~y 
—  g  -\-y  =  o  is  the  differential  equation  we  are  interested  in. 

Perfectly  generally,  if  we  have  a  relation  which  involves  n  arbitrary 
constants,!  we  differentiate  this  expression  n  times,  thus  having  in 

*  The  study  of  partial  differential  equations  will  be  taken  up  in  Chapter  XII. 

t  It  is  implied,  of  course,  that  the  n  constants  are  essential ;  that  is,  that  they  cannot 
be  replaced  by  a  smaller  number.  For  example,  y  =  x  +  a-f-  b  really  involves  only 
one  essential  constant,  since  a  +  b  is  no  more  than  a  single  constant.  Again  aex+*  is 
no  more  general  than  aex. 


DIFFERENTIAL   EQUATIONS 


§3 


all  n  +  i  equations  from  which  to  eliminate  the  n  constants.  So  that 
a  relation  (or,  as  we  shall  henceforth  call  it,  a  primitive}  in  which 
n  arbitrary  constants  appear,  gives  rise  to  a  differential  equation  which 
involves  derivatives  of  as  high  an  order  as  the  n\\\.  This  process  is 
unambiguous  ;  hence  a  primitive  gives  rise  to  one  and  only  one 
differential  equation.  Without  going  into  a  rigorous  proof  of  the 
fact,  we  see  how  a  primitive  involving  n  arbitrary  constants  gives  rise 
to  a  differential  equation  of  the  nth  order. 

To  illustrate,  find  the  differential  equations  corresponding  to  the 
following  primitives  :  — 


.   Here  c\  and  c%  are  the  arbitrary  constants. 


+  o 


Ex.  1.  y  = 

Then  ^  = 
ax 


From  these  three  equations  we  must  eliminate   ^   and  **2.      Con 
sidering  them  as  three  homogeneous  equations  in  the  quantities  i, 
%  we  nave 


Jl 

~d~x 


«! 


Ex.  2.    (x  —  <r)2  +/  =  r2.     Here  c  is  the  arbitrary  constant.    Then 
—  o.       From    these    two    equations   we    must 


dx 


eliminate  c. 

Now          x  _  c  =  — 

we  have 


. 
dx 


Substituting  this  in  the  original  equation, 


§4        DIFFERENTIAL  EQUATIONS  AND  THEIR   SOLUTIONS 


Ex.  3.  y  = 

Ex.4.  (x- 

Ex.  5.  y  =  ctf*  -f  cz. 

Ex.  6.  /  +  c&  =  o. 

Ex.  7.  oc*=2cy  +  <?. 

4.  General,  Particular  Solution.  —  If  now  we  start  with  a  differ 
ential  equation,  its  solution  involving  the  maximum  number  of  arbi 
trary  constants  is  nothing  but  the  primitive  which  gives  rise  to  the 
differential  equation.  That  solution  cannot  contain  more  than  n 
arbitrary  constants,  by  the  theorem  in  §  3.  Besides,  it  must  contain 
as  many  as  n  ;  otherwise  it  would  be  the  primitive  of  a  lower  ordered 
equation. 

The  solution  involving  the  maximum  number  of  arbitrary  con 
stants  is  called  the  general  (or  complete]  solution.*  By  means  of  the 
general  existence  theorem  (§  70),  we  can  prove  the  following 
theorem  :  —  The  general  solution  of  an  ordinary  differential  equation 
of  the  nth  order  is  one  that  involves  n  arbitrary  constants. 

Attention  should  be  called  to  the  fact  that  although  the  general  solution  may 
assume  a  variety  of  forms,  all  of  these  give  the  same  relation  among  the  variables, 
so  that  there  is  actually  only  one  general  solution  ;  thus  it  is  readily  seen  that 
x&y*  =  C  is  a  solution  of  (7);  so  also  is  log x  -f  logy  =  C,  or  \ogxy  —  C. 
These,  obviously,  are  all  equivalent  to  saying  that  xy  is  constant.  The  unique 
ness  of  the  general  solution  is  part  of  the  existence  theorem. 

A  solution  which  is  derivable  from  the  general  solution  by  assign 
ing  fixed  values  to  the  arbitrary  constants  is  called  a  particular 
solution.  Thus,  y  =  cos  x  and  y  =  cos  x  —  sin  x  are  particular  solu 
tions  of — =|-f-  y  =  o. 

*  We  shall  see  later  that  there  may  be  solutions  which  are  distinct  from  the  general 
solution.  In  the  general  theory  of  Differential  Equations  the  existence  of  a  solution 
for  every  differential  equation  (under  certain  restrictions)  is  proved.  The  solution 
there  obtained  is  the  general  solution  referred  to  in  the  text. 


6  DIFFERENTIAL   EQUATIONS  §4 

As  mentioned  in  §  2,  the  problem  of  the  Differential  Equations  includes  that 
of  the  Integral  Calculus  as  a  special  case.  Thus,  in  the  latter  the  general 
problem  is  to  solve 

£='<*>•      ' 

This  is  only  a  special  case  of  the  problem  of  finding  the  solution  of  the  differ 
ential  equation  of  the  first  order  involving  two  variables, 


where  f(x,y}  may  be  a  function  of  both  the  variables.  We  speak  of  integrating 
or  solving  the  equation,  in  the  general  case,  and  at  times  refer  to  the  simpler 
problem  of  the  Integral  Calculus  as  performing  a  quadrature. 

A  function  of  the  independent  variable  will  be  said  to  be  an 
integral  ®i  the  equation  if,  on  equating  it  to  the  dependent  variable, 
we  have  a  solution.  We  have  a  general  or  particular  integral  accord 
ing  as  the  resulting  solution  is  general  or  particular. 

While  the  problem  of  finding  the  differential  equation  correspond 
ing  to  a  given  primitive  is  a  direct  one,  and  can  be  carried  out 
according  to  a  general  plan,  involving  simply  differentiation  and 
elimination,  that  of  finding  the  primitive  or  general  solution  of  a 
given  differential  equation,  like  most  inverse  problems,  cannot  be 
solved  by  any  general  method. 

In  the  following  chapters  we  shall  bring  out,  in  as  systematic  a 
manner  as  possible,  some  of  the  classes  of  equations  whose  solutions 
can  be  found. 

We  shall  understand  that  the  problem  of  the  Differential  Equations 
is  solved  when  we  have  reduced  it  to  one  of  quadratures,  that  is, 
to  a  mere  process  of  the  Integral  Calculus.  While  in  the  general 
theory  of  the  Calculus  it  is  proved  that  every  function  has  an 
integral,  it  may  not  be  possible  to  express  it.  In  such  cases  we  shall 
content  ourselves  by  simply  indicating  this  final  process. 


CHAPTER    II 

DIFFERENTIAL  EQUATIONS   OF  THE  FIRST   ORDER  AND 
THE   FIRST  DEGREE 

5.  Exact  Differential  Equation.  Integrating  Factor.  The  general 
type  of  an  equation  of  the  first  order  and  degree  is 

(1)  Mdx  +  Ndy  =  o, 

where  J/and  -A^are  functions  of  x  andjy. 

Making  use  of  the  theorem  that  every  differential  equation  has  a 
general  solution  (§  70),  this  equation  has  a  solution  containing  one 
arbitrary  constant.  Solving  for  the  constant,  the  solution  has  the  form 

(2)  u(x,y)=C. 

The  differential  equation  having  this  primitive  is  obviously 

du  ,    .  du  , 
—dx  +  —dy  =  o. 
dx  dy 

Since  this  must  be  the  same  equation  as  (i),  we  must   have  the 
corresponding  coefficients  proportional,  i.e. 

du  du 
dx  __dy 
M~JV' 

If  we  call  this  common  ratio  p,  (which  is,  at  most,  a  function  of  x 
and  y),  we  have 

du         ,,    du         ,, 
—  =  nM,  —  =pN. 

dx  dy 

So  that  p(Mdx  +  Ndy)  =  du. 


8  DIFFERENTIAL   EQUATIONS  §5 

We  shall  speak  of  an  expression  which  is  the  differential  of  a 
function  of  one  or  more  variables  as  an  exact  differential.  Thus, 
p(Mdx  +  Ndy)  is  such,  since  it  is  the  differential  of  u.  We  shall 
further  speak  of  a  differential  equation  as  an  exact  differential  equa 
tion,  if,  when  all  the  terms  in  it  are  brought  to  one  side,  that  member 
is  an  exact  differential.  The  above  result  can  now  be  stated  as 
follows  :  Assuming  the  existence  of  the  general  solution  of  the  differ 
ential  equation  (i),  a  factor  ^  (x,  y)  exists  which,  when  introduced, 
will  make  the  equation  exact. 

This  factor  is  known  as  an  integrating  factor,  because,  as  we  shall 
see  (§  8),  when  our  equation  is  exact,  its  integration  can  be  effected 
readily. 

A  differential  equation  of  the  first  order  and  degree  has  an  indejinite  number 
of  integrating  factors. 

Suppose  n  to  be  an  integrating  factor.     Then 

ft  (Mdx  +  Ndy)  =  du  O,  y) 


where  the  sign  of  identity  =  means  that  pM  '=  —  and  p.N=  2^. 

dx  dy 

Now,  if  0(«)  is  any  continuous  function  of  u,  we  have 

«          dx  +       «      ^  d  =  du 


where 

du 


=  0(«),  or  ^(«)  =  ("0(«X«. 
J 


Hence  M0(«)  is  also  an  integrating  factor.  Since  0(«)  may  be  chosen  in 
an  indefinite  number  of  ways,  we  see  that  the  number  of  integrating  factors  is 
infinite.  [For  another  proof,  see  Ex.,  §  7;  also  §  80.]  Thus,  it  is  obvious  by 

inspection  that  x  dy  —  y  dx=o  has  —  for  an  integrating  factor.     We  have,  actu- 

oc1 

ally,  *<*y~y<l*=d  (A.     Here  p  =  1,  u  =y-.     Then  -<t>(y-\  will  be  an  in- 
x*  \x)  x11  x  x1     \x  j 

tegrating    factor.     In    particular,    —  —   or    —    is  an   integrating   factor   giving 

x^y          xy 

&  _  *L  Which  is  d(  log^  \  •    Similarly,  1  x-  or  ±  gives  d(  -  ^  \ 
y       x  \       x)  x1*  y*       y*  \     yl 


§§6-7         THE  FIRST  ORDER  AND  THE  FIRST  DEGREE  9 

6.  General  Plan  of  Solution.  —  Since  every  differential  equation  of 
the  first  order  and  degree  which  can  be  solved  by  elementary  means 
has  integrating  factors,  it  would  seem  natural  to  try  to  find  such  a 
factor  when  the  problem  of  solving  an  equation  of  this  type  arises. 
Practically,  this  is  not  always  possible  or  desirable.     In  the  following 
paragraphs  of  this  chapter  will  be  found  the  more  important  and  the 
more  frequently  occurring  classes  of  equations  of  the  first  order  and 
degree  which  can  be  solved  by  elementary  means  ;    and  it  will  be 
noticed  that  they  will  be  solved,  in  general,  either  by  finding  inte 
grating  factors  for  them  or  by  transforming  them  into  other  forms 
for  which  integrating  factors  are  known. 

7.  Condition  that  Equation  be  Exact.  —  If  the  equation  is  exact 
to  begin  with,  of  course,  no  integrating  factor  need  be  sought.     We 
must,  then,  find  the  necessary  and  sufficient  condition  for  exactness 
of  an  equation.     If 

(j\  Mdx  +  Ndy  =  o 

is  exact,  that  is,  if  M  dx  +  N  dy  is  the  differential  of  some  function  u, 
then       =  J/and—  =  W  and 


dx  dy 

\ 


- 

-T  ---  r—  > 

ay        dx 


d*u          d2u    * 

since  —  —  —  =  —   —  .    (2)  is,  then,  a  necessary  condition  for  exactness 
dx  dy      dy  dx 

of  the  equation. 

We  shall  now  prove  that  it  is  also  sufficient.  Even  more,  we  shall 
show  that  if  (2)  holds,  we  can  actually  find  a  function  u  such  that  its 
differential  is  Mdx  +  Ndy,  or,  what  is  the  same  thing,  such  that 

(3) 


, 
dx  dy 

*  Assuming  the  continuity  of  M  and  N,  and  the  existence  and  continuity  of  5— 


and         . 
d* 


10  DIFFERENTIAL   EQUATIONS  §7 

In  order  that  the  first  of  these  relations  (3)  should  hold,  we  must 
have 
(4)  u 

where  Y  is  a  function  of  y  only,  and  plays  the  role  of  a  constant 
of  integration,  since  y  is  considered  a  constant  in  the  process  of 
integration  involved  in  (4).  The  value  of  u  given  by  (4)  will 
satisfy  the  first  of  (3),  no  matter  what  function  of  y  Y  may  be.  In 
order  that  u  satisfy  the  second  of  (3)  we  must  have 

du       d    rx  ,,,     ,  dY      ,, 
—  =—  I   Mdx+-j-**N\ 

oy      dyj  ay 

that  is,  Kmust  satisfy  the  equation 


(5)  -  =  N-~      Mdx. 

dy  dy  J 

Since  the  left-hand  member  of  (5)  is  a  function  of  y  only,  the 
same  must  be  true  of  the  right-hand  member ;  that  is,  the  latter 
must  be  free  of  x,  or  in  other  words  it  must  be  a  constant  as  far  as 
x  is  concerned,  and  its  derivative  with  respect  to  x  must  be  zero. 

As  a  matter  of  fact,  that  derivative  is ,f    which   is   zero 

ox        dy 

because  of  (2).     We  can,  then,  find  Fto  satisfy  (5),  viz. 


and  the  resulting  value  of  u  in  (4)  will  satisfy  both  equations  (3). 

Ex.  Using  the  fact  that  (2)  is  the  necessary  and  sufficient  con 
dition  for  exactness,  prove  that  if  /x  is  an  integrating  factor,  such 
that  p(Mdx  4-  Ndy)=du,  /A  <£(*/)  is  also  an  integrating  factor. 

*  By  |  *  Mdx  we  mean  the  result  of  integrating  Mdx  considering  y  as  a  constant. 

Obviously  -~-  (    Mdx  =  Mt  since,  in  both  of  the  processes  involved,  v  is  con 
dor  J 


f 
sidered  constant. 


§8  THE   FIRST  ORDER   AND   THE   FIRST   DEGREE  II 

8.   Exact  Differential  Equations.  —  This  suggests  a  method  of  solv 
ing  an  exact  equation.     For,  in  this  case,  M  '  dx  +  N  dy  =  du  where 


So  that  the  general  solution  is 

(6)  C*Mdx  +  f  [V-  Y  f  XM  dx\dy  =  e. 

Expressed  in  words,  the  operations  involved  in  (6)  are  :  Integrate 
Mdx,  considering  y  as  a  constant,  thus  obtaining  f  *  M  dx.  Subtract 
ing  the  derivative  of  this,  with  respect  to  y,  from  N,  a  function  of  y 
only  is  left.  The  integral  of  this  function  plus  (  *  M  dx  is  the  left- 

hand  member  0f(6). 

f)  f*x 

Remark.  —  Frequently  N  --  I    M  dx  is  nothing  but  those  terms 

dyj 

of  N  free  of  x.  This  suggests  the  simple  rule  :  Integrate  M  dx,  con 
sidering  y  as  a  constant  ;  then  integrate  those  terms  in  N  dy  free  of  x. 
The  sum  of  these  equated  to  an  arbitrary  constant  is  the  general  solu 

tion.  This  rule  may  fail  because  f  *  Mdx  is  not  unique  as  far  as 
terms  involving  y  only  are  concerned.  Thus  J  *(x  +y)dx  may  be 

either  %x*  +  xy  or  %(x  +  y)2.  See  also  Ex.  3.  But  the  rule  is  found 
to  work  so  often  that  it  seems  worth  mentioning,  with  the  understand 
ing,  however,  that  when  it  is  employed,  the  result  be  redifferentiated 
to  see  whether  the  original  equation  is  obtained. 

As  an  exercise  let  the  student  show  that  the  general  solution  may 
also  be  obtained  in  the  form 


/ 

3  /2  xy  +  i\  i         d  (y  —  x\      , 

•r-f  —  -  -  )=  --  -  —  —  K         )  ;  hence  equation  is  exact. 
dy\      y       J          /      dx\  /    ) 

*2*y_±±jx  =  xz  +  £  a     i  is  the  only  term  in  ^V  free  of  x. 
v  v       v 


12  DIFFERENTIAL   EQUATIONS  $8 


'.  x  -\  ---  [-  logjy  =  c  is  the  general  solution. 


V    —  2  *  2  V    —  T 

Ex.2.  2X  2         * 


xf  —  x3 
Let  the  student  prove  that  this  is  exact  by  showing  that 

dM  =dN 

dy        dx 

dx  =  n/^b^  =  r*  _  r  * 

x3          J      x2  —  x2  J    x      J      *  — 


2 

At  first  sight  it  might  seem  that  no  term  of  N  is  free  of  x.     But 
writing  it  in  the  form  ^      ^  ~  x  ?  =  —  2-  --  h-,  we  see  that  there 


y 
is  such  a  term,  viz.  -• 

y 

:.  the  solution  is  log  x  +  \  log  (y2  —  x2)  4-  log  y  =  c, 
or  x2y2(y2  —  x2)=c. 

Ex.3.         dx       4-f1- 


Let  the  student  prove  that  this  is  exact. 

J    —    x       =  log  (x  +  yV  _f-  y)  ,  the    form   given    in    integral 
•  2       * 


tables. 

jV  contains  the  term  -  which  is  free  of  x.      Hence  if  the  rule  is 

y 

followed  blindly,  the  solution  would  seem  to  be 

log  (x  +  V.*2  +/)  +  log  y  =  c. 


§9  THE   FIR^T   ORDER   AND   THE   FIRST   DEGREE  13 


As  a  matter  of  fact,  the  solution  is  log  (x  +  V.*2  +/"')  =  c. 

/X  /7'Y' 

—  • 


=  log  (x  +  V*2  +  a2)  —  log  a. 

Naturally  in  tables  of  integrals  the  term  —log a  is  omitted,  as  that 
may  be  incorporated  in  the  constant  of  integration.  In  the  case  of 
the  problem  under  discussion  it  makes  a  big  difference  whether  this 
term  is  used  or  not.  If  it  is  used,  the  rule  gives  the  correct  result. 
But  if  the  form  of  the  integral  as  given  in  the  tables  is  used,  then 
the  rule  is  at  fault.  Hence  the  caution  given  in  the  remark  above 
should  be  heeded. 

Ex.4,    {y -\- x)  dx -\- x  dy  =  o. 

Ex.5.    (6*—  2y  +  i)dx-\-  (zy—  2X  —  $)dy  =  o. 


9.   Variables  Separated  or  Separable.     In  case  M  is  a  function  of  x 

only,  and  N  one  of  y  only,  the  relation =  — -  is  obviously  satisfied. 

dy        dx 

In  this  case  we  shall  say  the  variables  are  separated.      The  integral 
is,  of  course, 

\Mdx  +  \Ndy  —  c. 

Very  frequently,  the  variables  are  separable  by  inspection. 
Ex.  1.    sec  x  cos2 y  dx  —  cos  x  sin  y  dy  =  o. 

Hence  «£*<&_ 

cos  x 

or  sec2jr  dx  —  tan  y  sec  y  dy  =  o. 

The  general  solution  is  tan*  —  secjy  =c. 

*  Here is  an  integrating  factor,  found  by  inspection. 

cos  x  cos2;/ 


14  DIFFERENTIAL   EQUATIONS  §  10 

Ex.2,    (i  -\-x)^dx  —  x3dy  =  o. 

Ex.  3.    2(1  —f)xydx-\-  (i  -\-xz)(i  -i-y2)  dy  —  o. 

Ex.4,   sin  x  cos2^  dx  -+-  cos2  x  dy  =  o. 

10.  Homogeneous  Equations.  A  widely  occurring  class  of  equations 
where  the  variables  can  be  separated,  not  by  inspection,  but  by  a 
simple  transformation  of  variables,  is  that  in  which  M  and  N  are 
homogeneous  functions  of  x  and  y,  and  of  the  same  degree.  Then 

M 

—  is  a  homogeneous  function  of  degree  zero,  and  is  therefore  a  func- 

r     V* 

tion  of  J—  . 

x 

Our  equation  can  now  be  written 


dx          N 

Let  ^  be  a  new  variable,  say  v. 
oc 

Then  y  =  vx,  Q  =  v  +  x—  =  F(v\ 

dx  dx 

— *_.* 

F  (v)  —  v      x 

and  our  variables  are  separated. 

Integrating  this,  and  then  replacing  v  by  its  value  in  terms  of  x 
and  y,  we  have  the  general  solution. 

*  A  convenient  definition  of  a  homogeneous  function  of  x  and^  of  degree  r  is,  that  if 
in  the  function  we  replace  x  and  y  by  tx  and  ty  respectively,  where  /  is  anything  we 
please,  the  result  will  be  the  original  function  multiplied  by  t*.  (This  definition  can  be 
extended  at  once  to  a  function  of  any  number  of  variables.  It  is  obviously  consistent 
with  the  old  definition  of  homogeneity  of  polynomials.)  This  definition  can  be  form 
ulated  thus  :  If/ (x,y)  is  a  homogeneous  function  of  degree  r,  then /(to,  ty)  —  trf(x,  y). 

If  we  pot /»:—,  we  have  f  (  i,^  )=—  /  (x,y),  or  f(x,y)  =  xrf  ( i,¥  J.     When 


§10  THE   FIRST  ORDER  AND  THE   FIRST   DEGREE  15 

Ex.1.   \xf+y\dx  —  xdy  =  Q. 


Put  y  =  vx,  dy  =  vdx  +  x  dv.     Then 

x(e°  +  v)  dx  —  xv  dx  —  x2  dv  =  o, 

dx      dv 
or  ---  -  =  o.     Integrating,  we  have 

_y 

log  x  +  e~v  =  c,  or  log  x  -f  e~*  =  c. 
Ex.  2.  2  *>  +  3  /  —  (x3  +  2  #/)  ^  =  o. 

Put  7  =  w,  ^  =  e;  +  ^-     Then 
i  +  2  z;2          ^r 


dv       v  dv       dx 
or  --  1  —  —  ^  —  -  —  =  o.     Integrating,  we  have 

log  v  +  \  log  (i  +  z;2)  —  log  x  =  £, 
or  log  v*  +  log  (  i  4-  z>2)  —  log  x2  =  2  £, 

^(i+z'2) 
or  2  —  t-=^*=s^j  whence 


Ex.  3.   (/  —  xy)  dx  +  x*dy  =  o. 

Ex.4.  2xy+f-x*^  =  o. 
dx 

Ex.  5.  fdx  -\-  x*dy  —  o. 

Ex.  6.  f  x  -\-y  cos  -\dx-x  cos  -  ^y=o. 


-  :   ' 

1  6  DIFFERENTIAL   EQUATIONS  §  il 

11.   Equations  in  which  M  and  N  are  Linear  but  not  Homogeneous. 

If  M  and  TV  are  both  of  the  first  degree  but  are  not  homogeneous, 
we  can,  by  a  very  simple  transformation,  make  them  homogeneous. 
Suppose  our  equation  to  be  of  the  form 

fax  +  btf  +  *i)  dx  +  (a^x  +  b^y  +  c^)  dy  =  o. 

Put  x  =  x'  +  a,  y  =  y'  -f-  /?•     The  equation  becomes 
(<ii*'+  ttf'  +  W+  W  +  fi)  <&'  +  («**'  +  *&'  +  <*&  +  ^  +  ^2)  ^'  =  O, 
Now  we  may  choose  a  and  /2  such  that 

«i«  +  ^i/S  +  ^  =  o, 
and  #  2a  -f  ^2/8  +  ^2  =  °  \ 


•f  \Cn   -       »C\  j    /, 

/.^.  if  we  put        ce=    12    —  g_l  ?  and  y8  = 

^1^2  —  ^2^1 

our  equation  takes  the  form 


where  the  coefficients  are  homogeneous. 


Ex.1. 

Putting  •#  =  .#'  +  2,  7  =7'  —  3,  this  becomes 

(4  x'  +  3/)  </*'  +  (x1  +  7')  ^  =  o. 

Now,  if  y  =  vx\  we  get 

i  -+-  v         ,    .  dx1 


or 


2  +  V        (2+  rf         X' 

.:  log  (2  +  »)  +  --!-  4-  log  jf  '  =  r, 
or  log  ^'(2  +  z;)  =  ^  --  -  —  ,  whence, 

2  -\-  V 


§12  THE  FIRST   ORDER   AND  THE  FIRST  DEGREE  17 

Passing  back  to  x  and  y,  this  becomes 


Ex.  2.     (4*—  y  +  2)  dx+  ( 

Remark. — This  method  breaks  down  in  case  a\  b-2  —  azb\  =  o.  But  in  this 
case  we  can  find  a  transformation  which  will  separate  the  variables  at  once.  For 
we  have  ^  =  —  =  k,  a  constant,  and  the  equation  takes  the  form 


*1 

+  <r2]  dy  —  o. 


If  now,  we  put  a\x  +  b\  y  —  t,  so  that  y  =    ~  a^x  ,  our  equation  takes  the  form 


(b\  —  aik}  t  +  b\c\  —  aicz 
where  the  variables  are  separated. 

£\.  3.     (T.X  +  y)dx  —  (^x  +  27  —  i)  dy  =  o. 

12.  Equations  of  the  Form  yf\(xy)  dx  +  xf2(xy)  dy  =  o.  Another 
class  of  equations  in  which  the  variables  can  be  separated  by  a  sim 
ple  transformation  may  be  mentioned.  The  general  type  of  such  an 
equation  is 

x  -h  xfz(xy)  dy=Q, 


where  \sy  f(xy)  we  mean  a  function  of  the  product  xy. 

•if 

If  we  put  xy  =  v,ory  =  -,  then  xdy  —  dv  —  v-dx,  and  our  equa- 

"  -  ~  x  x 

tion  becomes          -  fi(v)  dx  +/2(v)dv  —  v-f^(v}dx  =  o, 

oc  oc 

^7S  +  J~°> 

where  the  variables  are  separated. 
&  (T 


1  8  DIFFERENTIAL   EQUATIONS  §  13 

Ex.  1.     (y  +  2xyi  —  x2y*)  dx  +  2x2y  dy  —  Q. 

Putting  y  =  -,dy=  *  dv  ~2  v  dx  ,  our  equation  becomes 
x  x 

v  /  ^  j  x  dv  —  v  dx 

-(l  +  2V  —  V*)dx-{-2  XV  -  ^—  -  -  =  O, 

oc  oc> 


, 

i  —  zr       x 


Integrating,  we  get,  since  „  =  —  -  --  1  --  -  —  , 

i  —  v2      i  +  v      i—0 


i  —  0 

Replacing  v  by  its  value,  we  get 

x  +  x?y  = 
i  —  xy 

Ex.2.    (2^+3^/)^r  +  (^  + 

Ex.  3.    (y  +  ^r/)^r  +  (x  —  x2y)  dy  =  o. 

13.  Linear  Equations  of  the  First  Order.  A  linear  differential  equa 
tion  (of  any  order)  is  a  special  kind  of  equation  of  the  first  degree. 
It  is  not  only  of  the  first  degree  in  the  highest  derivative,  but  it  is 
of  the  first  degree  in  the  dependent  variable  and  all  its  derivatives. 
Thus,  the  general  type  of  a  linear  differential  equation  of  the  first 
order  is 


where  P  and  Q  are  any  functions  of  x  only. 


§  13  THE  FIRST  ORDER  AND  THE   FIRST   DEGREE  IQ 

An  integrating  factor  for  this  equation   is  readily  seen*   to   be 

A***,   since    —  (  yeSpdx)  =  e$pd  x(&  +  Py\     Introducing    this   factor, 
dx  \dx          j 

we  have  the  solution  by  means  of  a  single  quadrature,  thus 


=  CQ  e$Pdx  dx  +  <r. 


Ex.1.  - 
dx 

Here  el***  =  e$cotxdx  =  elogainx  =  sin  x  is  an  integrating  factor. 
Using  this  we  have 


ysinx  =  J  smxsecxdx=  ( 


Hence  the  solution  is        y  sin  x  =  —  log  cos  x  +  ct 
or  y  sin  x  =  log  sec  x  +  e. 


Ex.2.   # 


*  We  shall  see  later  (§  80)  that  this  form  of  integrating  factor  arises  by  a  perfectly 
general  method.  Excepting  for  pedagogic  reasons,  that  method  could  be  given  at  this 
point,  without  assuming  anything  to  prevent  our  using  it  in  this  connection. 

f  While  the  above  method  of  solving  a  linear  differential  equation  of  the  first  order 
is  undoubtedly  the  simplest  in  both  theory  and  practice,  and  besides  has  the  advantage 
of  being  readily  retained  in  mind,  it  may  be  well  to  call  attention  to  a  second  method, 
which  is  part  of  a  general  method  applicable  to  linear  equations  of  any  order  (sec 
$§  S3.  59). 

Let  y=ylv.       Then    •-  =y\       +     r  •  »;   and  the  equation   becomes  ^i       + 


•£:+  Py\  )  v  =  Q.     If  we  choose  yl  so  that  -£±  +  Pyl  =  o,  we  get  y1  =  e~$P**  and 

the  equation  in  v  becomes  —  =  Q  e\P**t  whence  v  =   (  Q  eff**  dx  +  c;    and  we  have 
dx  } 

finally;/  =  e-fpd*  C  Q  e$P*c  dx  +  <*-J>*«. 


20  DIFFERENTIAL   EQUATIONS  §  14  j 

Putting  this 
unity),  we  get 


Putting  this  in  the  proper  form  (where  the  coefficient  of  -^  'is 


Cpdx  =  C(- 

/.  an  integrating  factor  is  e  Io=x+x,  or  x  ex. 
Using  this,  we  get  yxe*  =  I  e2x  dx  =  -  e2*  +  c 


EX.3.         __        =  (*+!)'. 
dx      x+i 

Ex.4.    (*  +  *»)£  +4*^=2. 


Ex.5. 

ax 

14.  Equations  Reducible  to  Linear  Equations.  —  At  times  an  obvi 
ous  transformation  will  change  an  equation  into  a  linear  one.  Such 
a  one,  of  frequent  occurrence,  is 


dx 

where,  as  before,  P  and   Q  are  functions  of  x  only,  and  n  is  any 
number. 

Dividing  by  yn  we  get 


. 

Let  y"n+1  =  v,  then  y-n^-  =  —  -  --  -,  and  our  equation  becomes 
dx      i  —  n  dx 


which  is  linear. 

*  This  is  known  as  Bernoulli's  Equation,  after  James  Bernoulli  (1654-1705). 


§14  THE   FIRST  ORDER  AND   THE  FIRST   DEGREE  21 


Ex.1,     i- 


-\dy          i+x      i 

2*y   ~ 


=  *,  then_3   -|^=^. 


2  I  — 


=  ,  cj 

J  i  — 


x 


=  —  2  log  (  i  — 


.*.  an  integrating  factor  is 
and  we  have      v  —^ 


l+x+x*  f 

-^7,  or 


i_t    i  +  *  +  . 

2«/    I  —  X3  (i  —  x)' 


dx 


(I- 


whence 


(i  - 


or  y*  =  -  *  - 

4  i  -f  x  +x2      i  -f  x 

As   examples   of  other   cases  where   an  obvious  substitution  will 
transform  the  equation  into  a  linear  one,  consider  the  following : 

Ex.  2.    y*2  -f  xf  =  x.      [Put  /  =  v.~\ 
ctoc 

Ex.  3.    sin  y-2  -f-  sinjc  cos  y  =  sin  x.     [Put  cos  y  =  v.~\ 


Ex.  4. 


3  y  + 


:0. 


22  DIFFERENTIAL  EQUATIONS  §  15 

15.  Equations  of  the  Form  xry*  (my  dx  +  nxdy~)  +  xpy«  (/z/  dx  + 
vx  dy)  —  o. 

Since  d  (*a/)  =  *a--/  -1  (^  </#  -f  bx  dy),  it  is  easily  seen  that  if 
we  start  with  any  expression  xrf  (my  dx  -f  nx  dy),  we  can  make  it 
exact  by  multiplying  it  by  x°-yfi,  provided 

<:/«  =  a  +  r-f-  i,  and  en  =  {$  +  s  -f  I, 
where  c  is  any  number.     As  a  matter  of  fact 
xcm-r-yn-°-lxrf(my  dx  +  nxdy)  =  xcm~lfn-l(my  dx  +  nx  dy) 

=  -d(x°mfn). 

If  c  =  o,  this  term  must  be  replaced  by  d  log  xmyn. 

The  object  of  introducing  the  undetermined  quantity  c  is  to  enable 
us  to  find  an  integrating  factor  for  an  equation  of  the  form 

xrf  (my  dx  -\-  nx  dy)  +  x^y*  (p/y  dx  -f  vx  dy)  =  o. 

Just  as  the  factor  xcm~"r~lycn~8~l  will  render  the  first  set  of  terms 
exact,  so  x^~p~lyyv~ff~  l  will  render  the  second  set  of  terms  exacl. 
In  order  to  find  an  integrating  factor  for  the  equation,  we  must  so 
determine  c  and  y  that  these  two  factors  are  one  and  the  same,  i.e. 
we  must  have 

cm  —  r—  i  —yt^  —  p—  i, 
en  —  s  —  i  =  yv  —  o-  —  i. 

These  two  equations  are  sufficient,  in  general,  to  determine  c  and  y* 

Ex.1.   x*y($ydx+2xdy)+x'2(4ydx  +  2xdy)  =  o. 

Here,          ^  =  3,  n  —  2,  r=4,  s=i,  /A  =  4,  "  =  3>  p  =  2,  o-  =  o. 

3'-5=4y-3>  2*-2  =  3y-i. 

From  these  we  have,  c  —  2,  7=1.  Hence  the  integrating  factor 
is  xy*.  Introducing  this,  we  get  the  solution 


=  flt  or  xy  +  2 

*  If  mv  —  «/*,  the  equation  reduces  to  the  simple  form  mydx—nxdy  =  o. 


§  16  THE   FIRST   ORDER   AND   THE  FIRST   DEGREE 

Ex.  2.    y2  (3  y  dx  —  6  jc  ^/v)  —  a;  (jy  ^r  —  2  #  ^)  =  o. 
Ex.  3.    (2  ^  —  y2)  dx  —  (2  ^4  4-  tfy)  </y  =  o. 


16.  Integrating  Factors  by  Inspection.  —  Integrating  factors  can 
frequently  be  found  by  inspection  on  closely  examining  the  terms 
entering  in  the  equation.*  Of  course  no  general  rule  for  this  can 
be  given.  A  commonly  occurring  group  of  terms  is  x  dy  —  y  dx. 

,-,,.                   x  dy  —  y  dx        x  dy  —  y  dx        dy     dx         x  dy  —  y  dx 
This  suggests  — -^ ,or      y  ./      ,or-f-~,or       *     •> 


* 


the  factors  being  respectively  — ,  — ,  — ,  — ••     So  that  —  is  an 

x2   /    xy    x2±y  x2 

integrating  factor  for  an  expression  of  the  form  xdy  —  y  dx  +f(x)  dx, 
while  -^  may  be  used  for  one  of  the  form  x  dy  —  y  dx  +f(y)  dy,  and 

—  for  x  dy  —y  dx  -\-f(xy)(x  dy  -\-  y  dx),    and   — -2  for  x  dy  —  y  dx 

->rf(xi±f)(xdx±ydy).     Other  combinations  will  occur  to  one  in 
actual  practice. 

Ex.  1.    (/  -  xy)  dx  +  x*dy  =  o.  (Ex.  3,  §  10.) 

Writing  this          f  dx  —  x  (y  dx  —  x  dy)  =  o, 

we  see  that  — -  is  an  integrating  factor.     Using  this,  we  get 
xy 

dx__ydx  —  xdy_ 
whence  log  x  —  -  =  c. 


*  An  illustration  of  this  we  had  in  §  9,  where  by  the  introduction  of  a  factor  the 
Variables  were  separated  and  the  equation  thus  rendered  exact. 


24  DIFFERENTIAL   EQUATIONS  §  17 

x  dy  —  y  dx  i 

Writing  this   — •;      J  f     ^  =  xdy}  we  see  at  once  that  —   is  an 

oc 


integrating  factor.     Using  this,  we  get     X  •;      •*  —  =  </y ;   whence 


Ex.  3.    (#  +  jv)  ^r  —  (A:  —  j)^/y  =  o. 

Writing  this  xdx+ydy+ydx  —  xdy**&i  we  see   at  once   that 


-—  -  -  is  an  integrating  factor. 
or  -j-  y1 


Ex.4.    (o?-\-f)dx  — 

Ex.  5.    (x  —  f)  dx-\-  2  xy  dy  =  o. 

Ex.  6.   jc  ^  —  ^  ^r  =  (V2  +/)  ^r. 


17.   Other  Forms  for  which  Integrating  Factors  can  be  Found.  —  In 

applying  the  test  for  exactness  (§  7),  we  find  the  value  of  -  --- 

dy       ox 

If  this  turns  out  to  be  zero,  the  equation  is  exact.  If  not,  it  may 
contain  either  M  or  JVas  a  factor.  By  the  general  method  of  §  80 
(already  referred  to  in  a  footnote,  §  13),  it  will  be  seen  that  if 


y         X  is  a  function  of  x  only,  say  /i(^),  then  e^(x)dx  is  an  inte- 

dN_d_M 
grating  factor,  and  if  -  y    is  a  function  of  y  only,  say/2(/), 


then  eff&w  is  an  integrating  factor. 


§i;  THE  FIRST  ORDER  AND  THE   FIRST  DEGREE  2$ 

It  may  be  interesting  to  call  attention  to  the  fact  that  this  method  of  §  80  also 

informs  us  that  -  is  an  integrating  factor  in  case  M  and  N  are  homoge- 

xM  +  yN 

neous  and  of  the  same  degree,  and  that  —  —  is  an  integrating  factor  if 


These  two  classes  of  equations  were  considered  in  §§  10  and  12,  where  we 
found  transformations  that  separated  the  variables.  Leibnitz  (1646-1716)  and 
his  school  endeavored  to  solve  all  equations  of  the  first  order  and  degree  by  in 
troducing  new  variables  which  become  separable  in  the  transformed  equation. 
Euler  (1707-1783)  and  his  school  tried  to  solve  all  equations  of  the  first  order 
and  degree  by  finding  integrating  factors.  As  a  matter  of  fact,  these  are  fre 
quently  spoken  of  as  Euler  factors  or  multipliers.  But  the  idea  of  an  integrat 
ing  factor  seems  to  be  due  to  a  contemporary  of  his,  Clairaut  (1713-1765). 
Now  it  is  interesting  to  note  that  just  as  every  differential  equation  of  the  first 
order  and  degree  has  an  integrating  factor,  in  general,  so  it  can  be  proved  f 
that  by  a  proper  change  of  variables  every  such  equation  can  be  transformed  into 
one  in  which  the  variables  are  separable.  But  in  actual  practice  it  would  be 
awkward  and  difficult  to  carry  out  this  method  in  all  cases,  just  as  it  would  be 
inadvisable  to  insist  upon  finding  an  integrating  factor  in  every  instance. 

These  two  classes  of  equations  are  of  particular  interest  as  affording  examples 
of  cases  where  both  the  general  methods  of  solution  can  be  readily  applied. 

Ex.  1.    (3  x2  +  6  xy  +  3  /)  <&  +  (2  *"  +  3  xy)  dy  =  o. 

dN 

-        '.  e  *  =  x  is  an  integrating  factor. 


N  x 

Ex.2.    2xdx 

dN 

-  —  =  i.     .*.  e$dy  =  ey  is  an  integrating  factor. 
M 

*  These  methods  cease  to  apply  \i  xM-\-yN=Q\n  the  first  case,  and  if  xM—yN=o 
in  the  second.  But  in  either  of  these  cases,  the  solution  of  the  equation  is  effected 
directly  with  ease.  For  if  xM+yN  '  =  o,  the  equation  takes  the  form  —  =  —  —  =  ^, 
and  its  solution  is  ^  =  c\  on  the  other  hand,  when  xM—yN=  o,  the  equation  takes 

the  form  _^  =  —  —  =  —  ^  ,  and  the  solution  is  xy  =  c. 

dx          N          x 
t  See  Lie,  Di/erentialgleichungen,  Chapter  6,  §  5  ;  also  the  author's  Lie  Theory,  \  20. 


26  DIFFERENTIAL   EQUATIONS  §  18 


Ex.  3.    (j4  -f  2  y)  dx  +  (ay8  -f-  2  jv4  —  4  #)  dy  =  o. 
Ex.  4.    (x?y  -/)  <£:  +  (/*  -x*)dy  =  o. 

Ex.  5.   Solve   examples  of  §§  10  and  12  by  the  method  of  this 
paragraph. 

Ex.  6.    (y2  —  xz  +  2  mxy]dx  -f  (wy2  —  w#2  —  2  ^ry)  ^  =  o. 


18.  Transformation  of  Variables.  —  In  case  the  equation  to  be 
integrated  does  not  come  under  any  of  the  heads  treated  in  this 
chapter,  it  is  possible,  at  times,  to  reduce  it  to  one  of  them  by  a 
transformation.  No  general  rule  for  doing  this  can  be  formulated. 
The  form  of  the  equation  must  suggest  the  transformation  to  be 
tried.  The  following  examples  will  illustrate. 


Ex.  1.   x  dy  —y  dx  +  2  x^y  dx  —  x*dx  =  o. 

Here  xdy—ydx  suggests  the  transformation  -  =  #.     Making  this 
transformation,  our  equation  reduces  to 

-j-  4-  2  xv  —  x,  which  is  linear. 
dx 

An  integrating  factor  is  e^Zxdx  or  e"\ 


=  C 


Hence  the  general  solution  is 


Ex.  2.    (x+y)  dy  —  dx  =  Q. 

Here  x+y  suggests  the  transformation  x+y  =  v.     Making  this, 
our  equation  reduces  to 


v  dv— i 
in  which  the  variables  are  separable  at  once,  so  that  we  have 

,        v  dv        ,         dv 
dx  —  —   —  —  dv -. 

v+i  v+i 


§18  THE   FIRST   ORDER  AND  THE  FIRST  DEGREE  2? 

Integrating,  we  have  x  —  v  —  log  (v  4-  i)  +  c, 
i.e.  x  =  x+y  —  log  (.*+,)>+  1)  +  *, 

or  log  (x+y+i)=y  +  c. 

This  can  also  be  integrated  as  follows  : 
Writing  it  in  the  form 

dx  _ 

-  —  x  —yt 

dy 

we  see  it  is  linear,  considering  y  as  the  independent  variable. 
An  integrating  factor  is  «—  W»  =  e~  v- 

Hence  xe~y  —  \ye~ydy  =  —ye~y  —  e~v  +  c, 

or  x  -f  y  +  i  =  c&i' 

Ex.3.   xdx-\-y  dy+y  dx  —  xdy  =  o.     (Ex.  3,  §  16.) 

V 

Here  x  dx  +  y  dy  suggests  x*  +  y2,  iftafaydx  —  xdy  suggests  -  .   This 

x 

combination  suggests  the  transformation  x2  -\-y2  =  r2,  2  =  tan  0  ;    or, 

x 

what  is  the  same  thing,  x  =  r  cos  0,  y  =  r  sin  0.     Then 
x  dx  -{-y  dy  =  rdrt 


dx  =  cos  0  dr  —  r  sin 

dy  =  sin  0  dr  -f-  r  cos  6  dO. 


Our  equation  then  becomes      * 

dr 
- 

and  the  solution  is 


log  r—  6  =  c,  or  log  -y/^+y  —  tan"1  -=  c. 


28  DIFFERENTIAL   EQUATIONS  §  I* 

Ex.  4.  x—  —  ay  +  by*  =  cxZa.  *     This  special  form  is  characterized 
dx 

by  the  fact  that,  when  the  first  term  is  x-^L,  the  coefficient  of  y  is  the 

cloc 

negative  of  half  the  exponent  of  x  in  the  right-hand  member. 
Putting  y  =  xav,  the  equation  becomes 


dv          dx 
or 


in  which  the  variables  are  separated. 

19.  Summary.  —  In  actual  practice,  when  the  equation  M dx  -f- 
JVdy  =  o  is  to  be  integrated,  we  proceed  as  follows  : 

By  inspection  we  can  tell  when 

i°    the  variables  are  separated  or  readily  separable  (§  9), 
2°   J/and  A"  are  homogeneous  and  of  the  same  degree  (§§  10,  17), 
3°    the  equation  is  linear  or  directly  reducible  to  one  that  is  (§§  13, 14), 
4°   J/and  ./Vare  linear  but  not  homogeneous  (§  n), 
5°   M=yMxy)9N=xft(xy)  (§§  12,  17), 

6°   the  equation  is  of  the  form  xry*  (my  dx  +  nx  dy)  +  x*y  (py  dx  + 
vxtfy)  =  o  (§  15). 

If,  on  inspection  (and  with  a  little  practice  this  inspection  can  be 
made  very  rapidly),  the  equation  does  not  come  under  any  of  these 
heads,  apply  the  test  for  an  exact  differential  equation.!  It  may 
happen  that 

*  This  is  a  special  form  of  Riccati's  equation  (see  §73). 

f  It  may  be  possible  to  recognize  by  inspection  that  an  equation  is  exact.  In  such 
case  proceed  at  once  to  integrate.  Or  an  integrating  factor  may  be  obvious  by  inspec 
tion  (see  10°  below).  The  general  plan  is  to  recognize,  as  promptly  as  possible,  the 
general  head  under  which  any  particular  equation  comes.  In  this  summary  and  those 
of  succeeding  chapters  the  various  possible  methods  are  arranged  in  the  order  of  the 
ease  of  application  of  the  test  as  to  whether  any  particular  method  applies. 


: 


i9 


THE  FIRST  ORDER  AND  THE  FIRST  DEGREE 


29 


dy 


If  none  of  these  cases  arise,  it  may  be  possible  to 
10°    find  an  integrating  factor  by  inspection  (§16),  or  to 
it0   find  some  transformation  that  will  bring  the  equation  under  one 

of  the  above  heads  (§  18). 
12°   As  a  final  resort  the  methods  of  §§  80,  25,  and  72  may  be  tried 


Ex.    1.      x  Vi  —  /  dx  +  y  Vi  —  x2  dy  =  o. 


2  dy  —  o. 


Ex.    3.          _ 

dx 


Ex.    4.     (y-x)2-=i.     [Put  y  —  x  =  V]. 


ctoc 


Ex.    5.      x     - 


Ex.  6.  (i  —  x)  y  dx  +  (i  —y)  x  dy  =  o. 

Ex.  7.  (_y  —  jc)  dy  +y  dx  =  o. 

Ex.  8.  x  dy  —  y  dx  = 

Ex.  9.      r  +  «-- 


Ex.  10.     x  dy  —y  dx  =  V^2  —  y2  dx. 

Ex.11, 

Ex.12,     (x— 


30  DIFFERENTIAL   EQUATIONS  §  19 

Ex.13. 


Ex.  14.     (  i  —  x2)  -+  —  xy  =  axy2. 

(IOC 

1    Ex.  15.     xy2  (3  y  dx  +  x  dy]  —  (2  y  dx  —  x  dy)  =  o. 

Ex.16,      (i  -h*2)  %y+y  =  tan-1x. 
&v 

Ex.  17.     (5  Xy-$f)dx  +  (3*'  ->jxyy)dy  =  o. 
Ex.  18.     —  -h  V  cos  #  =  -sin  2  tf. 

dk  2 

Ex.  19.  (xy*  -\-y)  dx  —  x  dy  =  o. 

Ex.  20.  (i  —  x)  y  dx  —  (i  +  y)x  dy  =  o. 

Ex.  21.  3  ^  ^r  +  (x3  +  jc3/)  ^v  =  o. 

Ex.  22.  (jc2  +/)  (xdx  +  ydy)=.  (x2  +y*  +  x)(xdy  —y  dx). 

Ex.23.  (2X  +  $y—i)dx  +  (2x  +  3y  —  s)dy  =  o. 

Ex.  24.  (jy3  —  2  ^)  ^r  +  (2  J^2  —  Jt:3)^  =  o. 

Ex.  25.  (2  jc3/  -y)dx  +  (2  x2f  -  x)dy  =  o. 

Ex.  26.  O2  +/)  (x  dx  +y  dy)  +  (i  +  *2  +/)^(jV  dx  —  x  dy)  =  o. 

~ 

Ex.  27.        i  +  <?y    ^r  +  <?2/f  i  --    rt    =  o. 


Ex.  28.     .#  dy  +  (jV  —  72  log  #)  </^  =  o. 

Ex.  29.    (jc3/  -f  jc2/  +  xf  +y)  dx  +  (jc4/  —  ^3/  —  ^  +  ^)  dy  =  o. 

Ex.  30.     (2  vSy  —  J?)  ^  H-  y  ^-'  =  o. 


CHAPTER    III 
APPLICATIONS 

20.  Differential  Equation  of  a  Family  of  Curves. —  Differential  equa 
tions  arise  in  certain  problems  in  Geometry  and  the  physical  sciences. 
For  example,  if  we  let  x  and  y  be  the  rectangular  coordinates  of  a 
point  in  the  plane,  any  relation  among  these,  say  </>(#,  y)  =  o,  repre 
sents  some  curve,  and  the  value  of  -2  at  any  point  of  this  curve  is 

dx 

the  slope  of  the  tangent  at  that  point. 

Starting  with  a  relation  that  involves  an  arbitrary  constant  rationally 

(i)  4>(x,y,c)  =  o, 

we  have  a  family  of  curves,  one  curve  corresponding  to  each  value 
of  c.     The  differential  equation  corresponding  to  (i)*,  say 


w  ;(*,»*)- 


is  of  the  first  order.     Since  we  can  obtain  from  it  the  value  of  -7 

dx 

corresponding  to  any  pair  of  values  of  x  and  y,  we  see  that  (2)  defines 
the  slope  of  the  tangent  of  that  curve  of  the  family  (i)  which  passes 
through  any  chosen  point  (x,  y).  In  case  (i)  is  of  the  second  degree 
in  c,  we  have,  excluding  exceptional  cases,  two  curves  of  the  family 
passing  through  any  point,  for  to  each  pair  of  values  of  x  and  y 
correspond  two  values  of  c  which  are  distinct,  in  general.  If,  now, 
we  turn  our  attention  to  the  differential  equation  (2),  it  must  give  us 

dy 
two  values  of  --  for  each  pair  of  values  of  x  and  y,  in  general,  since 

*  The  curves  defined  by  (i)  are  spoken  of  as  the  integral  curves  of  (2). 

31 


32  DIFFERENTIAL  EQUATIONS  §20 

two  distinct  curves  pass  through  this  point,  and,  excepting  in  the 

points  where  the  curves  are  in  contact,  their  tangents  will  be  distinct. 

If,    on  the  other  hand,  we  start  with  the  differential  equation  and 

dy 
suppose  that  it  is  of  the  second  degree  in  — ,  it  is  clear  that  since  at 

each  point  of  general  position,*  we  have  two  values  for  the  slope,  i.e. 
two  tangents  to  the  integral  curves,  we  must  have,  in  general,  two 
integral  curves  passing  through  each  point.  Hence  it  follows  that 
the  general  integral  involves  the  constant  of  integration  to  the  second 
degree.  (See  footnote  to  Ex.  3,  §  24.) 

Perfectly  generally,  we  can  prove,  by  entirely  analogous  reasoning, 
that  the  integral  of  a  differential  equation  of  the  first  order  and 
nth  degree  involves  the  constant  of  integration  to  the  nth  degree. 

Ex.  1.  Find  the  differential  equation  of  all  circles  through  the 
origin  with  their  centres  on  the  axis  of  x. 

The  equation  of  this  family  of  circles  is  evidently  x2  -\-f  —  2ax  —  o. 
Here  a  enters  to  the  first  degree. 

dy 
Differentiating,  we  have  x  -\-y  —  —  a  =  o. 

Eliminating  a,  we  get 

2  xy  -2  +  x2  —  y2  =  o,  which  is  of  the  first  degree. 
dx 

[As  an  exercise,  the  student  should  integrate  this.] 

Ex.  2.  Find  the  differential  equation  of  all  circles  of  fixed  radius 
r  and  with  their  centers  on  the  axis  of  x. 

The  equation  of  this  family  of  circles  is  (x  —  a)2  +  f  —  t3.  Here 
a  enters  to  the  second  degree. 

Differentiating,  we  have,  (x  —  a)  -\-y-J  =  o. 

*  By  point  of  general  position,  we  mean  a  point  at  which  there  is  nothing  peculiar 
about  the  family  of  curves  (such  as  having  two  curves  tangent  to  each  other  there),  or 
about  any  of  the  curves  of  the  family  (such  as  a  double  point). 


§20  APPLICATIONS  33 

Eliminating  a,  we  get 

y2  (  —}  +y2  =  r2,  which  is  of  the  second  degree. 
\dx) 

Ex.  3.    Find  the  differential  equation  of  the  system  of  confocal 

conies  whose  axes  coincide  with  the  axes  of  coordinates. 

x2-      v2 

Hint.  —  Since  the  distance  of  the  focus  of  the  conic   —±-/-=i 

a2      b2 

from  the  center  is  V«2  T  b2  it  is   quite   clear   that   all  the   conies 

~  -f  -I—  =  i  have  the  same  foci,  no  matter  what  c  may  be.    Hence 

c      c  —  \ 

this  is  the  equation  of  a  system  of  confocal  conies  whose  foci  are 

at  the   points  (±  VX,  o).     Here  c  is  the  arbitrary  constant  to  be 
eliminated. 

Ex.  4.  Find  the  differential  equation  of  the  system  of  parabolas 
whose  foci  are  at  the  origin  of  coordinates  and  whose  axes  coincide 
with  the  axis  of  x. 

Ex.  5.  Find  the  differential  equation  of  the  family  of  straight  lines 
tangent  to  the  circle  X2  ,  ^  __  ^2 

Ex.  6.  Find  the  differential  equation  of  the  family  of  straight  lines 
the  sum  of  whose  intercepts  on  the  axes  is  a  constant. 

.     Ex.  7.    Find  the  differential  equation  of  the  family  of  nodal  cubics 

(y-0)2=2x(x-^\ 

each  curve  of  the  family  being  tangent  to  the  axis  of  y,  and  having 
its  node  at  the  point  (i,  a). 

Ex.  8.    Find  the  differential  equation  of  the  family  of  nodal  cubics 

/=  2x(x-a)2, 

each  curve  of  the  family  being  tangent  to  the  axis  of  y  at  the  origin, 
and  having  its  node  at  the  point  (a,  o). 

Remark. —  If  the  equation  of  the  family  of  curves  involves  more  than  one 
arbitrary  constant,  the  corresponding  differential  equation  will,  of  course,  be  of 
higher  order  than  the  first  (§  3). 


34  DIFFERENTIAL    EQUATIONS  §  21 

Ex.  9.  Find  the  differential  equation  of  all  circles  tangent  to  the 
axis  of  y. 

Ex.  10.  Find  the  differential  equation  of  all  central  conies  whose 
axes  coincide  with  the  axes  of  coordinates. 

Ex.  11.  Find  the  differential  equation  of  all  parabolas  whose  axes 
are  parallel  (a)  to  the  axis  of  x,  (b)  to  the  axis  of  y. 

Ex.  12.  Find  the  differential  equation  of  all  circles  of  the  same 
radius  r. 

21.  Geometrical  Problems  involving  the  Solution  of  Differential 
Equations.  —  Differential  equations  of  the  first  order  arise  and  must 
be  solved  in  geometrical  problems  where  the  curve  is  given  by  proper 
ties  whose  analytic  expression  involves  the  derivative  of  one  of  the 
coordinates  of  a  point  on  the  curve  with  respect  to  the  other.  An 
example  or  two  will  illustrate  : 

Ex.  1.  Find  the  most  general  kind  of  curve  such  that  the  tangent 
at  any  point  of  it  and  the  line  joining  that  point  with  the  origin 
(which  we  shall  call  the  radius  vector  to  that  point)  make  an  isos 
celes  triangle  with  the  axis  of  x,  the  latter  forming  the  base. 

The  tangent  of  the  angle  between  the  tangent  line  and  the  axis 

of  x  is  -^-,  while  that  of  the  angle  between  the  radius  vector  and  the 
dx 

y 
axis  of  x  is  -' 

A- 

Hence  we  must  have 


Integrating,  we  have 

xy  =  constant, 

which  is  evidently  an  equilateral  hyperbola. 


§21  APPLICATIONS  35 

Ex.  2.  Find  the  most  general  kind  of  curve  such  that  the  normal 
at  any  point  of  it  coincides,  in  direction,  with  the  radius  vector  to  that 
point. 

Since  the  slope  of  the  normal  is ,  we  have 

dy 

dx     y  j  , 

=-/-9  or  xdx  -\- yay  =  o. 

dy      x 

Integrating,  we  have 

x2  +y*  =  c,  a  constant ; 

this  is  evidently  a  circle. 

Since  the  differential  equation  is  of  the  first  degree,  a  single  value 
of  the  constant  corresponds  to  a  pair  of  values  of  x  and  y.  Geo 
metrically,  this  means  that  through  each  point  passes  one  curve  of 
the  family.  Thus,  if  x=i,  y=z,  then  ^=5.  That  is,  through 
the  point  (i,  2)  passes  the  circle  x*-}-y2  =  $,  and  this  is  the  only 
circle  of  the  family  which  does. 

From  the  above  simple  examples  the  general  method  of  procedure 
may  be  seen.  It  consists,  first,  in  expressing  analytically  the  given 
property-  of  the  curve  (this  gives  rise  to  a  differential  equation)  ; 
secondly,  we  must  solve  this  equation ;  and  finally,  we  must  interpret 
geometrically  the  result  obtained.* 

In  the  Differential  Calculus  those  properties  of  curves  involving 
differential  expressions  are  usually  studied,  and  their  knowledge  will 
be  presupposed  here.  For  purpose  of  convenient  reference,  the 
following  list  will  be  given : 

i°   Rectangular  coordinates 

(a)  -2-  is  the  slope  of  the  tangent  of  the  curve  at  the  point  (x,  y)  ; 

*  The  general  solution  of  the  differential  equation,  involving  an  arbitrary  constant, 
represents  an  infinity  of  curves.  If  we  know  a  point  through  which  the  curve  must 
pass,  or  if  in  any  other  way  the  conditions  of  the  problem  determine  the  constant  ol 
integration,  either  uniquely  or  ambiguously,  one  or  several  curves  of  the  family  alone 
fulfill  the  requirements. 


36  DIFFERENTIAL   EQUATIONS  §21 

(K\ £  is  the  slope  of  the  normal  at  (xy  y) ; 

dy 

(c)  Y—y  =  —(X—x)  is  the  equation  of  the  tangent  at  the  poin 

ClOC 

(x,  y),  X  and  Y  being  the  coordinates  of  any  point  on  the  line ; 

(d)  Y— y  = (X—  x)  is  the  equation  of  the  normal  at  (x,  y) 

dy 

(e)  x—y—  and 7  —  #—  are  the  intercepts  of  the  tangent  on  th( 

ay  ax 

axes ; 

(/)  x+y-^-  and  y  +  x —  are  the  intercepts  of  the  normal  on  thi 
dx  dy 

axes  : 


Vfdx\2  /          / 'dv\2 

i  +  (  —  )    and  x\\  i  +  (  -^  )  are  the  lengths  of  the  tangen 
\dyj  ^         \dxj 

from  the  point  of  contact  to  the  x  and  y  axes  respectively ; 

(h)  y\i  +  (^-]    and  x \  i  +  [  —  )    are  the  lengths  of  the  normal 

1  x 


from  the  point  on  the  curve  to  the  x  and  y  axes  respectively ; 

(i)  y  —  is  the  length  of  the  subtangent ; 
dy 

(j)   y^-  is  the  length  of  the  subnormal ; 
dx 


the  ele 


(k)   ds  =  -\/dy*  +  dy*  =  Jxi  +  ^Y  =  dy  ^|i  +(—  Yis 
ment  of  length  of  arc  ; 

(/)   y  dx  or  x  dy  is  the  element  of  area. 
2°    Polar  coordinates 

7/J 

(m)   tan  ^  =  p  —  ,  where  ^  is  the  angle  between  the  radius  vector 
dp 

and  the  part  of  the  tangent  to  the  curve  drawn  towards  the  initial 
line; 

(n)   T  =  0  +  \f/,  where  r  is  the  angle  which  the  tangent  makes  with 
the  initial  line  ; 


§  2i  APPLICATIONS  37 

(o)   p  tan  \1/  =  p2  —  is  the  length  of  the  polar  subtangent  ; 
dp 

(/)   pcotij/  =  -£  is  the  length  of  the  polar  subnormal  ; 


ds  =  V>2  +  A*1  =i  +  2=^  +  2  isthe 


element  of  length  of  arc  ; 

(r)   |  p2  di0  is  the  element  of  area  ; 

(V)  p  =  p  sin  i^  =  p2  —   is  the  length  of  the  perpendicular  from 
ds 

the  pole  to  the  tangent. 


P 

Ex.  3.  Determine  the  curves  such  that  the  normal  (from  the 
point  on  the  curve  to  the  axis  of  x)  varies  as  the  square  of  the 
ordinate.  In  particular  find  that  curve  which  cuts  the  axis  of  y  at 
right  angles. 

Using  (/i),  we  have  the  differential  equation  of  the  curve 

=    °r  r  +    =     °r 


Integrating,  we  get 
k 


or  y=J-f  cek*  +  -  e  ~kx  ],  a  family  of  catenaries. 

2k\  C  ) 

To  find  the  curve  of  the  family  which  cuts  the  axis  of  y  at  right 

angles,  we  must  find  that  value  of  c  for  which  -^  =  o    for    x  =  o. 

dx 

Now  \-£\      =-(<:—-).    This  equals  zero  for  c  —  ±  i  .     Hence  the 

\<tx)**        2\  C) 

equation  of  the  required  curve  is  ±y=  —  (ekx  4-  e~kx)  =-  cosh  kx. 

2k  k 


38  DIFFERENTIAL   EQUATIONS  §22 

Ex.  4.  Determine  the  curve  such  that  the  area  included  between 
an  arc  of  it,  a  fixed  ordinate,  a  variable  ordinate,  and  the  axis  of  x  is 
proportional  to  the  corresponding  arc. 

Using  (/&)  and  (/),we  have 


This  is  the  same  differential  equation  that  arose  in  Ex.  3.  Hence 
the  catenary  has  also  this  property. 

Ex.  5.  Find  the  curves  such  that  the  polar  subtangent  is  propor 
tional  to  the  radius  vector. 

Using  (o),  we  have  /o2^  =  kp,  or  ^  =  dB. 

dp  p 

Integrating,  we  have  pk  =  ce&,  a  family  of  spirals. 

Ex.  6.    Determine  the  curves  whose  subnormals  are  constant. 

Ex.  7.  Determine  the  curves  whose  subtangent  at  each  point 
equals  the  square  of  the  abscissa  at  that  point. 

Ex.  8.  Determine  the  curves  such  that  the  perpendicular  from  the 
origin  upon  the  tangent  is  equal  to  the  abscissa  of  the  point  of 
contact. 

Ex.  9.  Determine  the  curves  such  that  the  angle  between  the 
radius  vector  and  the  tangent  is  one-half  the  vectorial  angle. 

Ex.  10.  Determine  the  curves  whose  polar  subtangent  is  four 
times  the  polar  subnormal. 

22.  Orthogonal  Trajectories.  —  A  curve,  which  cuts  every  member 
of  a  family  of  curves  according  to  some  law,  is  called  a  trajectory 
of  the  family.  Thus,  it  may  cut  every  curve  at  a  constant  angle  ;  if, 
in  particular,  that  angle  is  a  right  angle,  the  trajectory  is  said  to  be 
orthogonal.  It  is  at  times  possible  to  find  a  second  family  such  that 


§  22  APPLICATIONS  39 

each  curve  of  the  one  family  is  cut  at  right  angles  by  every  curve 
of  the  other.      If  such  a  pair  of  families  of  curves  exists,  each  is 
said  to  be  a  set  of  orthogonal  trajectories  of  the  other. 
Starting  with  the  first  family  whose  equation  is 

(i)  4  (x,  y,  *)  =  o, 

we  find  the  corresponding  differential  equation 


dy 

which,  as  we  have  noted  before,  defines  —  ,  the  slope  of  the  tangent 

ctoc 

at  (x,  y)  to  the  curve  of  the  family  through  that  point.     Obviously 
(3) 


will  have  for  integral  curves  a  family  such  that  the  slope  of  the  tan 
gent  to  the  curve  through  the  point  (x,  y)  at  this  point  will  be  the 
negative  reciprocal  of  that  in  the  case  of  the  corresponding  curve  of 
(i)  ;  i.e.  wherever  a  curve  of  the  one  family  cuts  one  of  the  other 
their  tangents  are  at  right  angles,  and  therefore  the  curves  are  said  to 
be  at  right  angles  themselves. 

The  integral  of  (3)  will  then  be  the  equation  of  the  desired  family. 

Ex.  1.   Find  the  orthogonal  trajectories  of  a  family  of  concentric 
circles. 

Taking  the  common  center  as  the  origin,  the  equation  of  the  circles 

is  x2  +  y2  =  <r2,  and  their  differential  equation  is  x  +y~r  —  °- 

Hence  the  differential  equation  of  their  orthogonal  trajectories  is 

oc 

Integrating,  we  have 

y  =  ex, 

a  family  of  straight  lines  through  the  center  of  the  circles. 


dx  dx      dy 

x  —y —  =  o,  or ^  =  o. 

dy  x       y 


4O  DIFFERENTIAL   EQUATIONS  §22 

Ex.  2.  Find  the  orthogonal  trajectories  of  the  family  of  circles 
through  the  origin,  with  their  centers  on  the  axis  of  x. 

The  equation  of  this  family  of  circles  is  x2+y2  —  ex  =  o.  We 
have  seen  (Ex.  i,  §  20)  that  the  corresponding  differential  equa- 

dy 

tion  is  2  xy  -^-+ x2—y2  =  o.     Therefore,  the  differential  equation  of 
ctoc 

the  orthogonal  trajectories  is  2  xy—  —  x2+f  =  o. 

dy 

As  this  equation  is  obviously  derivable  from  the  other  by  inter 
changing  x  and  y}  its  integral  must  be  x~  +  y2  —  cy  =  o,  the  family  of 
circles  through  the  origin  with  their  centers  on  the  axis  of  y. 

[Let  the  student  verify  this  result  by  actually  integrating  the 
equation.] 

Ex.  3.  Show  that  a  family  of  confocal  central  conies  is  self- 
orthogonal. 

The  equation  of  such  a   family  of  conies  with  their  axes  taken 

v2         v2 
for  axes  of  coordinates  is  (Ex.  3,  §  20) 1 — *• — =  i,  and   its  differ' 

C        C  —  A 

ential  equation  is  xy(-2-\  +  (x2  —  y'2  —  A.) -2-  —  xy  =  o. 
\dx)  dx 

Since  this  is  left  unaltered,  when  we  replace  -2-  by we  see 

dx  dy 

that  the  family  of  curves  is  self-orthogonal.  As  a  matter  of  fact, 
it  is  well  known  that  a  system  of  confocal  central  conies  is  made  up 
of  ellipses  and  hyperbolas,  such  that  through  any  point  there  pass 
one  ellipse  and  one  hyperbola,  and  these  cut  each  other  at  right 
angles.  (See  Ex.  17.) 

Ex.  4.  Prove  that  the  family  of  parabolas  having  a  common  focus 
and  a  common  axis  is  self -orthogonal. 


§22  APPLICATIONS  41 

Ex.  5.     Prove  that  the  differential  equation  of  the  family  of  trajec 
tories  which  cut  the  integral  curves  off  I  —  ,  x,  y  )  =  o  at  an  angle  a  is 

\  ClOC  / 


—  ,*,y    =o.' 

I+tanaS 

In  particular  show  that  the  trajectories  which  cut  the  lines  y  =  cx 
at  a  constant  angle  a  are  the  logarithmic  spirals 

-  log  (x2  -\-y2)  -\-k  —  —  tan"1  - , 

2  //*-  X 

0 
or  r—  cem. 

Ex.  6.  Find  the  trajectories  which  cut  at  a  constant  angle  a  the 
circles  through  the  origin  with  their  centers  on  the  axis  of  x. 

Ex.  7.  Find  the  trajectories  which  cut  at  a  constant  angle  a 
(other  than  a  right  angle)  a  system  of  concentric  circles. 

Ex.  8.     Find     the     orthogonal    trajectories     of     the     parabolas 

Ex.  9.     Find  the  orthogonal  trajectories  of  the  hyperbolas  x2— f—c. 

Ex.  10.  Find  the  orthogonal  trajectories  of  the  similar  central 
conies  ax2  +  by*  =  c,  where  a  and  b  are  fixed  constants  and  c  the 
arbitrary  constant. 

In  polar  coordinates  the  equation  of  a  family  of  curves  will  be 

(if)  <£(/>,  0,  c)  =o; 

and  the  corresponding  differential  equation  will  be 


which  defines  —  at  each  point  (p,  6}. 
dp 

*  In  practice  it  will  be  simpler  to  replace  tan  a  by  a  single  letter,  say  m. 


42  DIFFERENTIAL   EQUATIONS  §22 

We  have  noted,  §  21  (m),  that,  if  \p  is  the  angle  between  the  radius 
vector  to  the  point  (p,  0)  and  the  tangent  at  that  point  drawn  in  a 

definite  direction,  tan  \j/  =  p  —     If  now,  if/'  is  the  angle  for  the  curve 

dp 

cutting   this   one   at   right   angles   at   the    point    (p,   0),   we    have 
^'  —  i^=  ±-,  or  \f/'  =  \l/±  -•     Hence 

2  2 


tan  if/'  =  —  cot  $  =  — 


Using  primed  letters  for  the  second  curve,  we  have  then 


At   the   point   of  intersection   of  the  two   curves  p1  =  p,  0'  =  0. 
Hence 


is  the  differential  equation  of  the   orthogonal   trajectories   of   (i'), 

since  the  value  of  p  —  given  by  it  equals  that  of  —  -  -£  given  by  (2'). 
dp  p  dd 

Note.  —  Frequently  6  is  taken  as  the  independent  variable.     In 

this  case  (2')  will   be  in   the   form  /(&,p,  0]=o,  and  (3')  will 

\dO         J 

then  be 


Ex.    11.    Find  the  orthogonal  trajectories  of  the  family  of  lemnis- 
cates  p2  =  c  cos  2  6. 

Differentiating  and  eliminating  c,  we  find  the  differential  equation 

of  the  family  to  be  -    ^  =  —  tan  26.     Hence  the  differential  equa 
tion  of  the  orthogonal  trajectories  is 


§  23  APPLICATIONS  43 

-P^  =  -tan20, 
dp 

or  -£=cot 

p 

Integrating,  we  find  p2  =  k  sin  2  0,  a  second  family  of  lemniscates 
whose  axis  makes  an  angle  of  45°  with  that  of  the  first  family. 

Ex.  12.    Find  the  orthogonal  trajectories  of  the  family  of  cardi- 
oids  p  —  <r(i  —  cos  0). 

Ex.  13.    Find  the  orthogonal  trajectories  of  the  family  of  logarith 
mic  spirals  p  =  e°0. 

Ex.  14.    Find  the  orthogonal  trajectories  of  the  family  of  curves 
pm  sin  mO  =  cm. 

Ex.  15.    Find  the  orthogonal  trajectories  of  the  family  of  curves 

P 

Ex.  16.    Find  the  orthogonal  trajectories  of  the  family  of  confocal 

and  coaxial  parabolas  p  = • 

i  -  cos  0 

Ex.  17.   Find  the  orthogonal  trajectories  of  the  family  of  confocal 

c1  —  A.2 

conies   p  = : ,  c  being  the  parameter,  A.  a  fixed   constant. 

c  —  A.  cos  6 

23.    Physical  Problems  giving  Rise  to  Differential  Equations.  — 

Problems  frequently  arise  in  Mechanics,  Electricity,  and  other 
branches  of  Physics  whose  solution  involves  the  solution  of  differen 
tial  equations.  As  a  knowledge  of  these  subjects  is  necessary  to 
understand  properly  the  problems  that  arise  in  them,  we  shall  restrict 
ourselves,  as  far  as  possible,  to  problems  involving  only  very  elemen 
tary  principles.  As  in  the  case  of  geometrical  problems,  the  mode 
of  procedure  is,  first  the  analytic  expression  of  the  given  data  of  the 


44  DIFFERENTIAL   EQUATIONS  §23 

problem,  (this  gives  rise  to  the  differential  equation)  ;  then  comes 
the  problem  of  solving  this  equation,  with  an  interpretation  of  the 
result.  Frequently  there  is  the  additional  step  of  fixing  the  value  of 
the  constant  of  integration  so  as  to  satisfy  the  requirements  of  the 
problem. 

The  following  examples  will  illustrate  : 

Ex.  1.  A  body  falls  vertically,  acted  upon  by  gravity  only.  If  it 
has  an  initial  velocity  z>0>  what  will  be  its  velocity  at  any  given  instant, 
and  what  will  be  the  distance  covered  in  any  given  period  of  time  ? 

The  motion  being  rectilinear,  a  single  coordinate,  x,  will  be  suf 
ficient  to  determine  the  position  of  the  body.  Suppose  the  position 
of  the  body  at  the  time  /=  o  to  be  ^0  (this  is  called  its  initial 
position). 

The  velocity,  which  we  shall  represent  by  v,  is  —  ,  and  the  accelera 

tion,  represented  by/,  is  —  •     In  case  gravity  acts  alone,  the  accelera- 

dt 
tion  is  constant,  and  this  constant  is  usually  designated  by  g.     We 

,  dv 

have  now  —  =  g. 

at 

Integrating  this  equation  we  get 

v=gt+c. 

When  /=  o,  we  have  given  v  =  v0.  .'.  c  —  v^  and  the  answer  to  our 
first  question  is  given  by 


To  find  the  position  of  the  body  at  any  instant  we  must  integrate 

dx 


The  general  solution  of  this  is 


§  23  APPLICATIONS  45 

Since  X  =  XQ  when  /=o,  we  must  have  C  =  XQ.     Hence  at  any 
instant  /,  x  =  \g?  +  vj+x^ 

and  the  distance  covered  in  the  period  /  is 


Ex.  2.  A  particle  descends  a  smooth  plane  making  an  angle  a  with 
the  horizontal  plane.  The  only  force  acting  is  gravity.  If  the  par 
ticle  starts  from  rest,  find  the  velocity  at  any  moment  /,  and  the  dis 
tance  traveled  in  the  time  /.  \_Hint.  —  Here  the  acceleration  is 
g  sin  a,  the  component  of  g  in  the  direction  of  the  motion.]  Prove 
that  if  a  particle  starts  at  rest  from  the  highest  point  of  a  vertical 
circle,  it  will  reach  any  other  point  of  the  circle  when  moving  along 
the  chord  to  that  point  in  the  same  time  it  would  take  to  drop  to  the 
lowest  point  of  the  circle.  —  TAIT  AND  STEELE,  Dynamics  of  a  Particle. 

Ex.  3.  A  particle  falls  through  a  resisting  medium  (such  as  air)  in 
which  the  resistance  is  proportional  to  the  square  of  the  velocity  ; 
what  is  its  motion? 

The  equation  of  motion  is  then 

*-,-,*. 

Here  the  variables  are  separable,  and  we  have 
dv 


g-ktf 
Putting  gk  =  r2,  this  becomes 


=  dt. 


g*  —  rV  g  +  rv     g—  rv 

Integrating        j    f-     —  H —     -  ]=  2   \    dt,  we  have* 

*  A  knowledge  of  hyperbolic  functions  enables  one  to  effect  the  integration  much 
more  expeditiously,  especially  if  v0  =  o. 


\    — "- =  —  tanh — 1 

I     p-2  _  f'^i/i        f 


dt      r 


Integrating  again,  we  have  x  —  x0  =  &-  log  cosh  rt. 


46  DIFFERENTIAL   EQUATIONS  §23 

-D°g 


or 


+rvQg-rv 


g-rv     g-rvQ 
Temporarily  put  the  constant  %     rV(*  =  c.     Then 


Since  v  =  —  ,  the  position  at  any  time  is  given  by 
dt 

y~x  &    S*t  ^2rf  _  j 

I    dx  =  x  —  XQ  =  £  I    —  —  -  dt,  where  c  must  finally  be  replaced 
*A0  rJ<>    ce     -\-  i 

by  its  value  in  terms  of  z>0.     If  the  body  falls  from  rest,  ?;0  =  o,  /.  c  =  i. 
[As  an  exercise,  the  student  may  carry  out  the  integration  indi 
cated  above.] 

Ex.  4.  The  acceleration  of  a  particle  moving  in  a  straight  line  is 
proportional  to  the  cube  of  the  velocity  and  in  the  opposite  direction 
from  the  latter.  Find  the  distance  passed  over  in  the  time  /,  the 
initial  velocity  being  VQ,  and  the  distance  being  measured  from  the 
initial  position  of  the  particle,  i.e.  x$  =  o. 


dt 


and 


Ex.  5.    Find  the  distance  passed  over  in  the  time  /,  if  the  accelera 
tion  is  proportional  to  the  velocity. 


23  APPLICATIONS  47 


Ex.  6.   One  of  the  important  equations  in  the  theory  of  electricity 

*£+*-*, 

where  /is  the  current,  Z  the  coefficient  of  self-induction  (a  constant), 
R  the  resistance  (a  constant),  and  E  the  electromotive  force,  which 
may  be  a  constant  (including  zero)  or  a  function  of  the  time.  This 
equation  is  linear  (§13).  Find  /  if  (a)  E=  o,  (b)  E  =  constant, 
(<:)  E  =  EQ  sin  to/,  (E0  and  w  being  constants),  (d)  E  =  any  function 
of  the  time,  say  E(t). 

Case  (c}  plays  such  an  important  role  in  the  Theory  of  Electricity 
that  its  solution  is  given  here  in  detail.  This  equation  arises  in  the 
case  of  alternating  currents,  where  the  electromotive  force  is  a  peri 
odic  function  of  the  time,  the  period  being  £5,  and  the  maximum 

O) 

value  of  the  electromotive  force  is  EQ. 

di  ,  A  .     EQ    . 
_-+7,=:-J>  smart 
at      L         L 


An  integrating  factor  is  ei?  . 

.-.  ie~L  =  —  °  Ce~L*  sin  utdt  +  C. 

Since    f>  sin  »tdt=a  sin  *»'-  *>cos«»'  ^, 

J  fl58  +  o)2 


r> 

—  sin  w/  —  to  cos  <o/ 


^  sin  to/  —  o>  Z  cos  w/ 


48  DIFFERENTIAL   EQUATIONS  §23 

EQ  R  sin  w/  —  toZ  cos  at    %   . 


U  '       ^  WZ  JL  ^ 

where  sin  <f>  =  —  _ ,  cos  d>  = 


i  =  —  —  sin  (w/1  — 


-«l 

The  term  ^<r  L  usually  becomes  negligible  after  a  very  short  inter 
val  of  time.  The  current  then  becomes  periodic  with  the  same  fre 
quency  as  the  electromotive  force.  But  the  two  are  not  in  the  same 
phase,  the  current  lagging  behind  by  the  angle  <£. 


CHAPTER   IV 

DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER  AND 
HIGHER  DEGREE  THAN  THE  FIRST 

24.  Equations  Solvable  for  p.  —  For  the  sake  of  simplicity  we  shall 
adopt  the  generally  accepted  notation  of  replacing  -*•  by/.  Our 
equation  may  then  be  written 

(1)  f(p,x,y)  =  °- 

If  this  equation  is  of  the  nth  degree,  we  may  look  upon  it  as  an 
algebraic  equation  of  the  nth  degree  in  p.  Let  its  roots  be/i(x,  y), 
fz(x,y)>  "',fn(x>y)>  then  the  equation  may  also  be  written 

(2)  \J  -/&,  7)]  [/  -/,(*,  y)-]  ...  [>  -/„(*,  y)-]  =  o. 

Consider  now  the  differential  equations  arising  on  equating  each  of 
these  factors  separately  to  zero.  If  we  can  integrate  each  of  these 
by  some  method  of  Chapter  II,  we  can  readily  get  the  general  solu 
tion  of  (i).  Let  the  solutions  of  the  separate  equations  arising  from 

(2)  be  <fo  (x,  y,  c)  =  o,  <£2  (x,  y,  c)  —  o,  -  •  -,  <£„  (x,  y,  c)  =  o.    It  is  quite 
clear  that 

(3)  <h(x,y,  c)<h(x,y,  c)4>*(x,y>  *)  ••'•  4>*(*,y,  f)  =  o* 

is  the  solution  of  (i).  For,  the  vanishing  of  the  left-hand  member  of 
(3)  means  the  vanishing  of  one  of  its  factors.  This  will  cause  one  of 
the  factors  of  (2)  to  vanish,  when  substituted  in  it,  and  consequently, 
the  original  equation  will  be  satisfied.  Besides,  a  constant  of  inte- 

*  If  one  of  the  <£'s  is  not  rational,  there  will  be  found  certain  other  irrational  ones 
which,  with  it,  form  a  set  of  conjugate  irrational  functions  whose  product  is  rational 
(see  Ex.  2  below).  The  result  of  rationalizing  any  0  of  the  set  is  the  same  as  this 
rational  product.  In  practice  it  is  frequently  desirable  to  make  use  of  this  fact.  The 
student  should  verify  this  fact  in  the  case  of  Ex.  2. 

49 


50  DIFFERENTIAL   EQUATIONS  §24 

gration  is  involved.*    We  see  that  the  constant  enters  to  the  same 
degree  in  (3)  that  /  does  in  (i).    (Theorem,  §  20.) 

Ex.   1.    f+(x+y)p  +  ^;  =  o, 
or  (/  +  *)(/+ JO  =  o. 

Integrating  -^  -f-  x  =  o,  we  get  2y  +  xz  =  c ; 

and  integrating  -2-  +y  =  o,  we  get  log  7  -f  x  =  k,  or  y  —  te~*. 
{too 

Hence  the  general  solution  is  (2y  +  x2  —  c)  (y  —  ce~*)  =  o. 
Ex.2.     xz—2      —  x  =  o. 


Solving  this  for  /,  we  get  /  =y  ±  v.     We  haye? 


consider  the  two  equations  xp  —  y  =  V.*2  4-  f 


and  xp—  y  =  —  V*2  -f/. 

=  A  may  be  written  ^y-ydx   =  & 

' 


on  integrating, 


[ 


*  Since  (3)  is  a  solution  by  having  each  of  its  factors  separately  satisfying  the  differ 
ential  equation  (i),  it  may  be  asked  why  we  use  the  same  constant  c  in  all  the  factors. 
If  we  look  upon  equation  (i)  as  equivalent  to  the  n  separate  equations  arising  on 
equating  each  of  the  factors  of  (2)  to  zero  (which,  in  fact,  it  is),  then  we  really  have 
«  equations  to  solve,  and  the  various  factors  of  (3),  involving  distinct  constants,  are  the 
solutions  of  these.  But  if  we  require  the  general  solution  of  (i)  to  be  given  as  a  single 
expression,  first,  there  is  no  room  for  more  than  one  constant  of  integration  (§  4),  ana 
then,  there  is  no  loss  in  using  the  single  constant  throughout,  from  the  very  fact  that  (3) 
is  a  solution  by  virtue  of  eagh  factor  separately  satisfying  the  equation. 


§24 


THE   FIRST  ORDER  AND   HIGHER   DEGREE 


Integrating  xp—y  —  —  vx?  +jv2,  we  get  in  a  like  manner 


log 


y 


—  i 


log  ,*  +  k,  or  - 


fy\* 

(  -  )  -  <r*  = 


o. 


Hence  the  solution  is 


or 


—  2  r>>  —  I  =  O. 


Ex.  3.  /+/=i. 
Here/  =  ± 


—    8.     Solution  of 


or  ^  =  sin  (#  +  <:)  ;  while  that  of 


dy 


=  =  dx  is  sin"1  y  =  x  +  c, 

=  dx  is  y  =  cos  (x  +  ^). 


Since  ^  is  perfectly  arbitrary,  either  one  of  these  is  sufficient  as  the 
general  solution  of  the  equation  ;  *  for  sin  (  x  -f  c  +  -  )  =  cos  (x  +  c). 


\ 

*  Since  c  enters  in  a  transcendental  way,  and  not  to  the  second  degree  in  the  solu 
tion^  =  sin  (x-\-c}t  or y=  cos  (x  -{-c},  we  seem  to  have  an  exception  to  the  rule  of 
§  20.  It  is  really  not  such.  That  rule  presupposes  that  the  constant  of  integration 
enters  algebraically.  We  can  make  our  solution  conform  to  the  rule  by  writing  it  in 
the  form  (sin— ^y  —  x  —  ^(cos-1^  —  x  —  c}  =  o.  As  a  matter  of  fact  the  rule  was 
based  on  the  fact  that  through  each  point  of  general  position  pass  two  integral  curves 
of  a  differential  equation  of  the  second  degree.  So  here,  of  the  infinite  number  of 
values  of  c  that  satisfy  y  =  sin  (x  +  c)  for  a  given  pair  of  values  of  x  and^,  only  two 
will  determine  distinct  curves.  A  more  elegant  form  than  the  one  above,  in  which  c 
enters  algebraically  and  to  the  second  degree,  may  be  gotten  thus :  Since  c  is  any 
number,  we  may  write  sin  c  =  —  —  ;  then  cos  c  =  — —  ;  whence,  remembering 

that  sin  (x  -f-  c)  =  cos  c  sin  x  +  sin  c  cos  x,  the  solution  takes  the  form 
y  —  cos  x  —  2  k  sin  x  -j-  &(y  +  cos  x}  =  o. 


52  DIFFERENTIAL   EQUATIONS  §25 

Ex.4.  (2xp-y)2=8x*. 

Ex.5.  (i+**)/=  i. 

Ex.  6.  /  -  (2  *+/)/  +  (*2  -/  +  2  */)/  -  (*2  -/)/  =  o. 


25.  Equations  Solvable  for  /.  —  If  the  equation  can  be  solved  for 
y,  the  following  method  will  frequently  be  found  useful.  Solving  for 
y,  we  have 

(4)  .y  =  ^  (*»/)• 

Differentiating  this  we  get 


a  differential  equation  which  is  really  of  the  second  order,  but  since 
y  is  no  longer  present,  it  may  be  looked  upon  as  an  equation  of  the 
first  order  in  the  variables  x  and  /.  It  may  happen  that  we  can  inte 
grate  this  equation.  Suppose  its  solution  to  be 

(6)  »(x,p)  =  c. 

Eliminating/  from  (4)  and  (6)  we  have 
(?)  <t>  (x,  y,  0  =  o> 

which  is  a  solution  of  (4)  ;  and,  since  it  involves  an  arbitrary  con 
stant,  it  is  the  general  solution. 

We  know  that  (7)  is  the  solution  of  (4)  from  the  following  consid 
erations  :  Since  (5)  is  the  derivative  of  (4),  every  solution  of  (4)  is 
a  solution  of  (5),  considered  as  an  equation  of  the  second  order  in  x 
and  y.  Since  (6)  is  a  solution  of  (5),  every  solution  of  (6)  looked 
upon  as  a  differential  equation  in  x  and  y  is  a  solution  of  (5).  (4) 
and  (6)  are  known  as  first  integrals  of  (5).  Since  (4)  contains  y  and 
(6)  does  not,  these  two  first  integrals  are  evidently  independent. 
Equation  (4)  has  an  infinite  number  of  integral  curves,*  and  so  has 

*  We  use  this  geometrical  mode  of  expression,  not  because  it  is  essential  to  the 
argument,  but  because  it  is  simpler. 


§25  THE   FIRST   ORDER   AND   HIGHER   DEGREE  53 

(6)  for  each  value  of  c.  To  find  the  curve  or  curves  common  to  the 
two  families  of  integral  curves  we  shall  have  to  find  the  equation  of 
the  locus  of  those  points  for  which  (4)  and  (6)  determine  the  same 
value  of/.*  But  this  locus  is  evidently  gotten  by  eliminating/  from 
(4)  and  (6).  For  each  value  of  c  we  get  thus  one  integral  curve  of 
(4).  Hence,  when  c  is  an  arbitrary  constant,  we  have  the  general 
solution. -j- 

Note.  —  This  method  applies  equally  well  to  equations  of  the  first  degree.     See 
Ex.  4. 

Remark.  — At  tinaes  it  is  easy  to  integrate   (6).    Doing  this,  a  relation 
(8)  4>  O,  y,  c,c'}  -  o, 

involving  two  constants,  results.  This  is  the  general  solution  of  (5)  and  must 
therefore  contain  that  of  (4)  for  some  relation  between  c  and  c' .  This  relation 
can  be  found  by  substituting  (8)  in  (4)  and  noting  what  condition  is  imposed, 
so  that  the  equation  be  satisfied.  In  actual  practice,  however,  this  method  will 
generally  not  be  as  desirable  as  the  one  given  above. 

Ex.  1.    2  px  — y  -f-  log/  =  o  ;  or 
(i)  y=2px 

Differentiating,  we  get 


V          pjdx' 

or  /  dx  +  2  x  dp  +  -  dp=  o. 

/ 

An  integrating  factor  is  seen,  by  inspection,  to  be  /.     Using  this, 

we  have 

/2  dx  -f-  2px  dp  +  dp=o. 

Integrating  we  have 

(2)  /*+/=* 

Eliminating/  between  (i)  and  (2),  we  have  the  required  result. 

*  As  already  mentioned  (§  20),  a  differential  equation  of  the  first  order  may  be 
looked  upon  as  defining/,  the  slope  of  the  integral  curve,  at  each  point  (x,y). 

t  Since  the  process  of  eliminating  p  from  (4)  and  (6)  may  introduce  extraneous 
factors,  and  errors  may  enter  in  other  ways,  it  is  desirable  to  test  the  result  (7) ,  by  find 
ing  out  whether  it  actually  satisfies  the  equation  (4) . 


54  DIFFERENTIAL   EQUATIONS  §25 

Remark.  —  In  this  case,  while  it  is  perfectly  possible  to  perform  this  elimina 
tion  [since  (2)  can  be  readily  solved  for  /,  and  the  latter  value  can  be  put  in 
(i)],  the  result  will  not  be  very  attractive  in  form.  It  is  simpler  to  say  that 
(i)  and  (2)  taken  together  constitute  the  solution,  in  that,  from  them,  we  can 
express  x  and  y  in  terms  of  /,  which  may  be  looked  upon  as  a  parameter.  Thus 
from  (2)  we  have 


and  .-.  y  =  ±A£ — ^  +  log  p. 


Such  parametric  representation  is  frequently  resorted  to;   thus,  for   example, 
the  parametric  equations  of  the  ellipse 


(~  x  =  a  cos  6, 
y  —  b  sin  6, 


where  8  is  the  eccentric  angle,  and  also  the  usually  adopted  equations  of  the 
cycloid 


(x  =  a  (0  —  sin  0),  | 
y  =  a  (i  —  cos  6}.  J 


Ex.  2.   4  xp*  +  2  xp  —y  =  o  ; 

or  y  =  2  xp  -\-  4  xp*. 

Differentiating,  we  have  on  collecting  terms, 


Neglecting  the  factor  4/-f  i,  whose  significance  we  shall  see  later 
(§  32,  note),  we  have,  integrating  the  other  factor,  xp2  =  $. 
Hence  the  solution  is  y  =  2k  V#  +  4&2,  or  putting  2k  =  c, 


y  =  c^ 

Rationalizing  this,  we  have 

(y  —  r  )2  =  c~x,  or  putting  <r2 


§26  THE   FIRST   ORDER   AND    HIGHER   DEGREE  55 

Ex.  3.  xp2  —  2  yp  —  x  =  o.      (Ex.  2,  §  24.) 

Ex.  4.  /  +  2  xy  =  x2  +/. 

Ex.  5.  y  =  —  xp  +  x*f. 

Ex.  6.  /2  +  2*/—  _y  =  o. 

26.  Equations  Solvable  for  x.  —  A  method,  entirely  analogous  to 
that  of  the  previous  paragraph,  can  be  deduced  in  case  the  equation 
can  be  solved  for  x.  Suppose  the  equation  in  the  form 


(9)  * 

Differentiating  with  respect  to  y  we  get 
,     ,  dx      i      80  .  86  dp 

(  IO)  —  =  —  =  --  --    ••"  • 

dy     p      dy      dp  dy 

Here  x  no  longer  appears,  and  we  may  look  upon  this  equation  as 
one  of  the  first  order  in  y  and  p.  If  we  can  integrate  this,  we  obtain 
a  relation  involving  an  arbitrary  constant, 


and  on  eliminating  p  between  (9)  and  (n)  we  have  the  general 
solution. 


Ex.1. 

Ex.  2.   cPyp*  —  2  xp  -f  y  =  o. 

Ex.  3.   xpz  —  zyp  —  x  =  o.     (Ex.  3,  §  25.) 

Ex.  4.  pz  —  4  xyp  +  8  y2  =  o. 

Ex.  5.  Find  the  family  of  curves  for  which  the  length  of  the  nor 
mal  (from  the  curve  to  the  axis  of  x)  is  equal  to  the  square  root  of 
the  length  of  its  intercept  on  the  axis  of  x. 


56  DIFFERENTIAL   EQUATIONS  §27 

27.    Clairaut's  Equation.*  —  If  the  equation  is  of  the  form 

where  f(p)  is  any  function  of/,  the  solution  is  gotten  so  readily  that 
especial  attention  should  be  given  to  this  form  of  the  equation,  in 
order  that  it  may  be  recognized  at  once.f 
Using  the  method  of  §  25,  we  have 


, 

or  *  =  o. 


Neglecting  the  factor  x  +/'(/),  which  involves  no  differential  ex 
pressions  (see  §  32,  note),  we  have 

-£  =  o,  whence  p  =  c. 

dx 

Putting  this  value  in  (i)  we  have 


which  is  the  general  solution  of  (i). 

Ex.1.    (Jx-y)*=jt+i. 

Solving  for  y,  y=px±  V/'J  -f-  i  .    • 

This  being  in  Clairaut's  form,  its  solution  is  known  at  once  to  be 


y  =  ex  ±  V ' r  +  i, 
or  (** ->)»  =  <*+ i. 

*  This  form  of  equation  is  named  after  Alexis  Claude  Clairaut  (1713-1765).  He 
was  the  first  to  apply  the  process  of  differentiation  ($$  25,  26)  to  the  solution  of  equa 
tions.  His  application  of  this  method  to  the  equation  that  bears  his  name  was  pub 
lished,  Histoire  de  r  Academie  des  Sciences  de  Paris,  1734. 

f  The  student  should  be  able  to  recognize  this  equation,  not  only  when  it  is  solved 
for.y,  as  it  is  in  (i).  Obviously,  what  characterizes  this  form  of  the  equation  is  that 
x  and  y  occur  only  in  the  combination  y—px.  Hence  any  function  of  y—  px  and 
/  equated  to  zero,  say  /  (y—px,  p}  =  o  is  a  Clairaut  equation,  and  its  solution  is 
/(y  —  ex,  c)  =  o.  (See  Ex.  I.) 


§27  THE   FIRST   ORDER   AND    HIGHER   DEGREE  57 

Ex,  2.    4  r»f  +  2  xp  -  i  =  o. 

Put  e2y  =  t,  then     /  —  -2-  =  —  --  ,  and  the  equation  becomes 
dx      2tdx 


Its  solution  is  /=  ex  +  r2,  or  <ry  =  <r.r  +  c1. 

Remark.  —  At  times,  as  in  the  case  above,  a  transformation  can  be 
found  to  simplify  very  materially  an  equation  which  will  not  yield 
directly  to  any  of  the  previous  methods.  Unfortunately  these  trans 
formations  are  not  always  obvious.  Experience,  and  frequently  that 
alone,  will  help  one  in  making  a  proper  choice. 

Ex.  3.    4  ^/  +  2  e-xp  -  e2x  =±  o. 


Put    ex  =  u,  e2y  =  vt  then  p  =  —  —  ,  and  the  equation  becomes 

2  v  du 

dv 

v=u 

du 
whence  its  solution  is        e~y  =  cc*  +  ^2. 

Note.  —  One's  first  impulse  would  be  to  try  e^  —  u  and  e^  —  v.     Our  equa 

tion  then  becomes  v=  2u—  -}-  qu  (  -~  )  •     This  is  not  in  Clairaut's  form;   but  it 
du  \dul 

can  be  integrated  (see  Ex.  2,  §  25),  so  that  this  transformation  is  also  effective. 

Ex.  4.    **/  +  (e-x  +  e*x}p  -  ^x  =  o. 

Ex.5.   x}fy2  —  fp  -}-  x  =  o.     (Let  x2  =  u,  y*  =  v.) 

Ex.  6.    ( 

(Let  x+y  =  u,  x2+y~  =  v.) 

Ex.  7.  y=2px  +//3.      (Let  /  =  v.) 

Ex.  8.    a*yf—  2  xp  +y  =  o.    (Ex.  2,  §  26.)    (Let  2  x  =  u,  y2  =  v.) 

Ex  9.     ^-2  =  ^22-^2. 


58  DIFFERENTIAL   EQUATIONS  §28 

28.  Summary.  —  Given  a  differential  equation  of  the  first  order 
and  higher  degree  than  the  first,  there  are  three  methods  *  which 
suggest  themselves,  to  be  tried  in  actual  practice  in  the  following 
order : 

i°  Solve  for  p,  and  then  solve  the  resulting  equations  of  the  first 
degree  (§  24). 

2°  Solve  for  y,  differentiate  with  respect  to  x,  integrate,  and 
eliminate  p  between  this  solution  and  original  equation  (§  25). 

3°  Solve  for  x,  differentiate  with  respect  to  yt  integrate,  and 
eliminate  p  between  this  solution  and  original  equation  (§  26). 

Clairaut's  form  (§  27)  has  been  given  special  prominence  in  this 
chapter  because  of  the  ease  of  finding  its  solution.  It  is,  of  course, 
solved  by  method  2°. 

If  none  of  the  above  methods  work,  a  substitution  must  be  sought 
to  bring  the 'equation  into  manageable  shape. 

There  are  certain  cases  when  we  can  tell  in  advance  that  some  or  all  of  these 
methods  work.  Here  the  difficulties  are  those  of  Algebra  or  the  Integral  Cal 
culus,  and  not  of  the  Differential  Equations.  For  example,  consider  the  follow 
ing  cases : 

(a)  If  the  equation  in  /  is  algebraic  and  all  the  coefficients  are  homogeneous 
and  of  the  same  degree  in  x  and/,  then,  on  dividing  by  the  leading  coefficient, 
all  the  coefficients  are  homogeneous  and  of  zero  degree.  Hence,  if  we  can  solve 
for/  (which  is  an  algebraic  process),  we  shall  find  /  as  homogeneous  functions 
of  x  and  y  of  degree  zero,  and  the  resulting  equations,  when  subjected  to  the 
transformation/  =  vx,  will  have  their  variables  separated  (§  10)  and  are  solvable 
by  quadratures. 

Again,  since  after   dividing  by  one  of  the  coefficients   of  the  equation   the 

equation  is  a  function  of  /  and  — ,  say/( /,  -  )  =  o,  if  we  solve  for  /,   (or  —  J» 
we  get/  =  •*V'(/)'     Differentiating,  we  have, 


where  the  variables  are  separated. 

*  It  is  almost  needless  to  remark  that  these  methods  are  not  mutually  exclusive. 
Two,  or  all  three,  methods  may  be  applicable  to  some  equations. 


§28  THE   FIRST   ORDER   AND   HIGHER   DEGREE  59 

Hence,  in  this  case  methods  i°  and  2°  both  work,  provided  we  can  solve  for 
/  and  for  y. 

(£)  If  x  is  absent,  so  that  the  equation  is  of  the  form/(/,  y)=  o,  solving  for/, 


or 


or 
dx 

Again,  solving  for  y,  we  get  /  =  !/'(/),  and  differentiating,  we  have/=  $'(!>}  •*-, 
rJUi!2_2?  —  dx,  where  the  variables  are  separated. 


Here  again  methods  i°  and  2°  both  work. 

(<:)  If  y  is  absent,  equation  is  /(/,  x)  =  o.  Let  the  student  show,  as  an 
exercise,  that  in  this  case  methods  i°  and  3°  both  work,  provided  we  can  solve 
for  /  and  for  x. 

(</)  If  the  equation  is  of  the  first  degree  in  x  and  y,  thus,  xf\  (/)  -\-yfi  (/) 
+/a  (/)  =  o  *,  it  is  readily  seen  that  method  2°  works.  For,  solving  for  y,  we 
have 

y 

Differentiating,  we  get 


Considering  /  as  the  independent  variable,  this  may  be  written, 

) 


•vhich  is  linear  and  can  be  solved  by  quadratures  (§  13). 

Ex.  1.  /(i  +/)  =  **. 

Ex.   2.  j/  =(#  —  ^)/  +  flr. 

Ex.  3.  ^3/  +  jc27/  +1=0. 

Ex.4.  3/x  —  6yp  +  x+  2y  =  o. 

Ex.  5.  y=/~(x+i). 

Ex.6,  (/je—  y)(py  +  x)  =a?p.     (Let  jc2  =  «,  /  =  ^. 

Ex.  7.  /+2/_ycot^=/. 

*  Claiiaut's  equation  is  a  special  case  of  this. 


60  DIFFERENTIAL   EQUATIONS  §28 

Ex.    8.    (I+,*2)/-  2xyp+f—  1=0. 
Ex.    9.    x-p*-2(xy+  2/)/+/  =  o. 

Ex.  10.  y  =  xp  +2^  •     (Let  ,#2  =  u,  /  =  z;.) 

j£ 

EX.  11.  *22  -  2 


Ex.  13.  Find  the  equation  of  the  curves  for  which  the  distance  of 
the  tangent  from  the  origin  varies  as  the  distance  of  the  point  of 
contact  from  the  origin. 

Ex.  14.  Find  .the  equation  of  the  curves  such  that  the  square  of 
the  length  of  arc  measured  from  a  fixed  point  is  a  constant  times 
the  ordinate  of  the  point.  (Let  the  constant  factor  be  4^.) 

Ex.  15.  Find  the  equation  of  the  curves  down  each  of  whose  tan 
gents  a  particle,  starting  from  rest,  will  slide  to  the  horizontal  axis 
in  the  same  time.  [As  in  Ex.  2,  §  23,  we  have  that  the  distance 

covered  in  the  time  /  equals  -gtz  sin  a.     Here  sin  a  =  —  and 


the  distance  covered  is  ^  ,  §  21,  (g).~\ 

P 


CHAPTER   V 
SINGULAR  SOLUTIONS 

29.  Envelopes.  —  We  have  noticed  before  that  <f>(x,  y,  ^)  =  o, 
where  c  is  an  arbitrary  constant,  represents  a  family  of  curves,  to 
each  value  of  c  corresponding  some  definite  curve  (provided  c  enters 
rationally,  which  we  shall  suppose  to  be  the  case  .throughout  this 
chapter).  So  that  if  we  pick  out  some  curve  corresponding  to  a 
definite  value  of  c,  we  can  suppose  our  attention  directed  to  the 
different  curves  corresponding  to  c  as  it  varies  continuously.* 

We  shall  be  interested  in  the  locus  of  the  ultimate  points  of  in 
tersection  of  each  curve  with  its  consecutive  one.  By  the  ultimate 
points  of  intersection  of  a  curve  with  its  consecutive  one  we  mean  the 
limiting  positions  of  the  points  of  intersection  of  a  curve  with  a 
neighboring  one  as  the  latter  approaches  coincidence  with  the  former. 
(Thus  in  the  case  of  the  family  of  circles  referred  to  in  the  footnote, 
the  ultimate  points  of  intersection  of  two  consecutive  curves  are  the 
extremities  of  the  diameter  perpendicular  to  the  axis  of  x.)  To 
find  the  equation  of  this  locus,  we  proceed  as  follows  :  If 

(1)  <j>(x,y,c)=o 

is  the  equation  of  a  curve  corresponding  to  some  chosen  value  of  c, 

(2)  <l>(x,y, 


*Thus,  for  example,  consider  the  family  of  circles  of  fixed  radius  r  whose  centers 
all  lie  on  the  axis  of  x  ;  their  equation  is  (x  —  t)2+>'2  =  f2.  When  c  =  o  we  have 
the  circle  whose  center  is  at  the  origin,  and  as  c  increases  we  get  circle  after  circle 
whose  center  is  (c,  o). 

61 


62 


DIFFERENTIAL   EQUATIONS 


will  be  the  equation  of  a  neighboring  curve,  A<r  being  a  finite  con 
stant  quantity,  different  from  zero.  To  find  the  points  of  intersection 
of  these  two  curves  we  have  to  solve  (i)  and  (2)  for  x  and  y,  or  we 
may  replace  (2)  by  a  constant  times  their  difference,  i.e.  by 


(3) 


=  0- 


To  obtain  the  ultimate  points  of  intersection  of  the  curve  with  its 
consecutive  one  we  combine  (i)  with  what  (3)  becomes  when  we  let 
A<r  approach  the  limit  o,  i.e.  with 


(4) 


d<l>(x,y,c) 
dc 


If  we  were  to  solve  (i)  and  (4),  we  should  actually  obtain  the  inter 
sections  of  the  curve  with  the  consecutive  one.     But  what  we  want 


is  the  locus  of  these  points,  for  all  values  of  c.  This  is  evidently 
gotten  by  eliminating  c  between  (i)  and  (4).  This  locus  is  known 
as  the  envelope  of  the  family  (i).  A  property  of  the  envelope  which 
we  shall  have  occasion  to  use  is  :  At  each  point  of  the  envelope  there 
is  one  curve  of  the  family  tangent  to  it.  This  is  immediately  obvious 
from  the  figure.  Suppose  (I),  (II),  (III)  are  three  curves  of  the 
family  which  ultimately  become  coincident,  a  becomes  an  ultimate 
point  of  intersection  of  (I)  and  (II),  and  b  of  (II)  and  (III).  Hence 
they  are  both  points  on  the  envelope,  and  the  line  joining  them  be 
comes  ultimately  a  tangent  to  the  envelope.  But  they  are  also  both 
on  the  curve  (II),  so  that  the  line  joining  them  also  becomes  a 


§30  SINGULAR   SOLUTIONS  63 

tangent  to  (II).  That  is  to  say,  (II)  is  tangent  to  the  envelope  at 
the  limiting  position  of  a  and  b.* 

Ex.    Find  the  envelope  of  the  family  of  circles  referred  to  in  the 
footnote,  page  61. 

30.   Singular  Solutions.  —  Suppose  now  that 
(i)  4>(x,y,c)=o 

is  the  solution  of  /  (  p,  x,  y)  —  o.  We  have  already  noted  that, 
looked  at  geometrically,  this  means  that  the  slope  of  the  tangent  at  a 
point  (x,y)  of  the  curve  of  the  family  defined  by  (i)  passing  through 
that  point  is  exactly  the  value  of/  given  by  /(/,  x,  y)  =  o  for  that 
pair  of  values  of  x  and  y.  But  we  have  just  seen  that  the  tangent 
at  any  point  of  the  envelope  of  the  family  of  integral  curves  coin 
cides  with  that  of  the  integral  curve  through  that  point.  It  follows, 
then,  that  the  equation  of  the  envelope  will  satisfy  the  differential 
equation,  and  is  consequently  a  solution.  Moreover,  since  the  en 
velope  is  usually  not  a  curve  of  this  family,  i.e.  its  equation  cannot 
be  gotten  from  (i)  by  assigning  a  definite  value  to  the  parameter, 
the  equation  of  this  envelope  is  a  solution,  distinct  from  the  general 
solution.  It  contains  no  arbitrary  constant,  and  is  not  a  particular 
solution.  It  is  known  as  the  singular  solution. 

Ex.  1.   y  =  px+-- 
P 

This  being  Clairaut's  equation,  its  solution  is 


or  x 

*This  theorem  ceases  to  hold  in  case  the  limiting  position  of  a  is  a  singular  point 
on  (II),  such  as  a  double  point  or  cusp.  See  Fig.  2,  §  33.  While,  as  we  shall  see  (§  33), 
if  each  curve  of  the  family  has  a  singular  point,  the  locus  of  these  satisfies  the  geomet 
rical  as  well  as  analytic  requirements  for  an  envelope,  it  is  usually  customary  to  apply 
the  term  envelope  to  that  part  of  the  locus  of  ultimate  points  of  intersection  of  the 
curves  of  the  family  which  has  a  curve  of  the  family  tangent  to  it  at  each  point. 


64  DIFFERENTIAL   EQUATIONS  §3 

Differentiating  with  respect  to  <r,  we  have 

2  ex  —y  =  o. 
Eliminating  c,  we  get  y2  =  4  x,  which  is  the  singular  solution. 

Ex.  2.     xp*  —  2  yp  —  x  =  o. 

This  is  the  equation  of  Ex.   2,  §  24.     We  saw  there  that  its  solu- 

tionis  «y-*gi-i.-« 

Differentiating  with  respect  to  r,  we  have 

r.r2—  ^  =  0. 

Eliminating  c,  we  have  the  singular  solution 


31.    Discriminant.  —  If  f(z)   is  a  polynomial  of  the  nth  degree, 


we  have  by  Taylor's  theorem 
f(a  +  /0  =/(fl 


where  /'  («  )  =  =  ^oo-1  +  («  -  i)  r,  «-»  +  . 


-  ^(«~   0 


,w-3 


§31  SINGULAR   SOLUTIONS  65 

Putting  h  =  z  —  a,  we  have 
/(*)  =/(«)'+/'  (a)  (*-* 

From  this  it  is  obvious  that  if  a  is  a  root  of  f(z),  i-e.  if/(0)  = o, 
/(z)  contains  z  —  a  as  a.  factor ;  and  conversely,  in  order  that  /(s) 
should  contain  the  factor  z — •  a  we  must  have/"(#)  =  o.  Similarly,  if 
a  is  a  double  root,  (z  —  #)2  is  a  factor  of  /(z),  so  that  we  must  have 
f(a)  =  o  and  ./'(«)  =  °  >  conversely,  if /(a)  =  o  and  f\ct)  =  o,f(z) 
will  contain  (z  —  a)'2  as  a  factor,  and  a  will  be  a  double  root.*  The 
necessary  and  sufficient  condition,  then,  that  f(z)  have  a  repeated 
root  is  that  /(z)  and  /'(z)  have  a  root,  say  a,  or  the  corresponding 
factor  z  —  a,  in  common.  This  condition  obviously  depends  upon  the 
coefficients  of/ (2).  That  rational,  entire  function  of  the  coefficients 
whose  vanishing  expresses  the  necessary  and  sufficient  condition  that 
/(z)  shall  have  a  repeated  root  is  called  the  discriminant  off(z).  It 
can  readily  be  shown  to  be  the  product  of  the  squares  of  the  differ 
ences  of  the  various  roots  of  the  equation  (multiplied  by  a  power  of 
<TO  to  avoid  fractions).  It  may  be  calculated  in  various  ways  :  The 
process  of  finding  the  greatest  common  divisor  will  show  when  /(z) 
and/' (2)  have  a  common  factor  z  —  a.  But  this  process  is  apt  to 
introduce  extraneous  factors.  A  better  way  is  to  eliminate  z  from 
/(z)  and  f(z),  (or  better  still,  from  nf(z)  —  zf(z)  and  /'(z)  which 
are  both  of  degree  n—  i).  The  result  of  this  elimination  is  a  relation 
among  the  coefficients  of  f(z),  which  expresses  the  condition  that 
f(z)  =o  and  f'(z)  =o  can  be  satisfied  simultaneously.  If  all  the 
terms  of  this  relation  are  brought  over  to  one  side  of  the  equation, 
and  the  expression  is  cleared  of  fractions  and  radicals,  we  have  evi 
dently  the  discriminant  equated  to  zero.  We  shall  call  this  the 
discriminant  relation.  Various  methods  for  eliminating  the  variable 

*  In  an  entirely  analogous  manner  it  can  be  seen  very  readily  that  f(a)  =  o, 
f'(a)  =  o,  /"(a)  =  o,  •••,/(r~1)(a)  =  o,  is  the  necessary  and  sufficient  condition  that 
a  be  an  r-fold  rppt, 


66  DIFFERENTIAL   EQUATIONS  §  32 

from  two  polynomials  involving  it  are  given  in  the  Theory  of  Equa 
tions.     It  will  be  sufficient  to  recall  here  that  the  discriminant  of  the 

quadratic  az~  +  bz  +  c  is 

b-—$ac, 

while  that  of  the  cubic  az*  +  bz"  +  cz  +  d  is 

P<*  +  18  abed-  4  a<*  -  4  £V  -'27  «Vf. 

In  the  case  of  $  (#,  j,  <r)  =  o,  looked  upon  as  an  equation  in  c,  the 
coefficients  are  functions  of  x  andjy.  It  may  be  possible  to  find 
values  of  x  and  y,  such  that  this  equation  shall  have  equal  roots. 
Geometrically,  this  amounts  to  saying  that  there  may  be  points 
through  which  there  pass  a  smaller  number  of  integral  curves  than 
usual ;  for  to  each  value  of  c  there  corresponds  a  distinct  integral 
curve.  The  discriminant  relation,  in  this  case,  is  the  locus  of  such 
points. 

32.  Singular  Solution  Obtained  directly  from  the  Differential 
Equation.  —  Since  the  problem  of  finding  the  equation  of  the  envelope 
of  <J>  (x,  y,  c)  —  o  is  identical  with  that  of  finding  the  discriminant 
relation,  we  see  that  through  each  point  of  the  envelope  there  pass 
a  smaller  number  of  integral  curves  than  through  points  of  general 
position  in  the  plane ;  that  is,  at  least  two  of  the  integral  curves 
through  each  point  of  the  envelope  coincide.  (Thus,  in  the  case  of 
the  family  of  circles  already  referred  to,  through  any  point  of  the 
envelope,  it  is  readily  seen,  only  one  circle  passes,  instead  of  two.) 

Since  there  is  at  least  one  less  curve  passing  through  a  point  of 
the  envelope,  there  will  be  at  least  one  less  tangent  to  the  curves 
through  such  a  point.  Hence  for  points  along  the  envelope,  the 
differential  equation/ (/,  x,  y)  =  o,  which  defines  the  slopes  of  the  tan 
gents  to  the  integral  curves  through  the  point  (x,  y),  will  have  at  least 
two  of  its  roots  equal,  i.e.  for  points  along  the  envelope,/(/,  x,y)  =  o 

and     -/ VA  x*y)  _  0  are  simultaneously  satisfied.     As  a  consequence,, 


§32  SINGULAR  SOLUTIONS  6? 

the  result  of  eliminating  p  between  these  two  equations  will  give  us 
the  equation  of  the  envelope  and,  therefore,  the  singular  solution, 
whenever  there  is  one.  See  §  33. 

Note. —  In  the  previous  chapter  we  came  across  certain  factors,  in  the  course 
of  solving  equations,  which,  while  they  would  have  led  to  solutions,  did  not  con 
tain  arbitrary  constants,  and  were  therefore  neglected  at  that  time.  It  will  now 
be  understood  that  these  factors  usually  lead  to  singular  solutions.  Thus,  in 
the  case  of  an  equation  in  Clairaut's  form  (§  27),  (i)  y  =  px  +  /(/),  the  neg 
lected  factor  is  (2)  x  +/'(/)  =  o.  But  this  is  exactly  the  derivative  of  (i) 
with  respect  to/.  So  that  if  we  eliminate  p  between  (i)  and  (2),  we  get  the 
singular  solution.  Again,  in  Ex.  2  (§  25),  we  neglected  the  factor  4/+  i  =  o. 
Eliminating  p  between  this  and  the  original  equation,  we  have  x  +  4y  =  o, 
which  is  a  singular  solution  of  the  equation,  but  not  the  whole  singular  solu 
tion.  Both  the/-  and  <:- discriminant  relations  are  x(x  +  4^)  =  o.  This  illus 
trates  the  fact  that  the  appearance  of  such  a  factor  in  the  course  of  the  work  im 
plies  a  singular  solution,  but  it  need  not  always  appear  when  a  singular  solution 
exists.  In  other  words,  this  is  not  the  way  to  look  for  singular  solutions, 
although,  in  actual  practice,  it  is  advisable  to  examine  these  factors  and  see  to 
what  they  lead. 

Remark.  —  From  the  fact  that  two  roots  of  an  equation  can  be 
equal  only  in  case  there  are  as  many  as  two  roots,  no  singular  solu 
tion  can  exist  in  the  case  of  equations  of  the  first  order  and  degree. 
But  it  may,  and  not  infrequently  does,  happen,  that  equations  of  a 
higher  degree  than  the  first  have  no  singular  solutions. 

Let  the  student,  as  an  exercise,  prove  that  a  differential  equation  of  the  first 
order  and  higher  degree  than  the  first,  which  is  decomposable  into  factors  linear 
in/  and  rational  in  x  and/,  cannot  have  singular  solutions. 

It  may  be  further  remarked  that  at  times  the  singular  solution 
gives  rise  to  a  result  that  is  much  more  interesting  than  that  arising 
from  the  general  solution. 

For  example,  let  us  ask  for  that  curve  which  has  the  property  of 
having  the  length  of  its  tangent  intercepted  by  the  coordinate  axes  a 
constant  /. 


68  DIFFERENTIAL   EQUATIONS  §31 

From  formulae  (<?),§  21,  we  have 


or 


Since  this  is  in  Clairaut's  form  (§  27),  the  general  solution,  when 
solved  for^,  is  seen  at  once  to  be 


This  represents  a  family  of  lines  whose  length  intercepted  by  the 
axes  is  /.  The  curve  that  we  are  actually  interested  in  is  obtained 
when  we  look  for  the  singular  solution.  This  is  gotten  by  finding 
either  the  /-  or  the  ^-discriminant  relation  (the  two  being  identical 
in  the  case  of  an  equation  in  Clairaut's  form).  It  is 


which  is  a  hypocycloid  of  four  cusps. 

Ex.  1.    Find  the  curve  for  which  the  product  of  the  perpendiculars 
drawn  from  two  fixed  points  to  any  tangent  is  constant. 

Ex.  2.    Find  the  curve  whose  tangents  are  all  equidistant  from  the 
origin. 

Ex.  3.    Find  the  curve  for  which  the  area  enclosed  between  the 
tangent  and  the  coordinate  axes  is  a2. 

Ex.  4.    Find  the  curve  such  that  the  sum  of  the  intercepts  of  the 
tangent  on  the  coordinate  axes  is  a  constant. 


§33  SINGULAR   SOLUTIONS  69 

Ex.  5.   Integrate  the  following  equations  and  examine  for  singular 

solutions  : 

xY  -  2  (xy  -2)p  +/  =  o, 


o. 


33.  Extraneous  Loci.  —  We  have  noticed  that  the  ^-discriminant 
relation  is  the  equation  of  the  locus  of  points  through  which  a 
smaller  number  of  curves  pass  than  ordinarily.  Now,  if  an  integral 
curve  has  a  double  point,  at  that  point  there  will  be  two  branches  of 
the  curve.  Since  there  are  only  n  values  of  /  (if  the  differential 
equation  is  of  the  nth  degree)  there  is  only  room  for  n  —  2  other  curves 
through  this  point.  Hence  this  point  must  be  on  the  locus  of  the 
c-  discriminant  relation.  And  if  there  are  an  infinity  of  integral 
curves  having  double  points,  or  nodes  as  they  are  sometimes  called, 
the  locus  of  these  points  (known  as  the  nodal  locus)  will  be  given  by 


FIG.  2  FIG.  3 


the  r-discriminant  relation.  Excepting  in  the  unusual  case  where 
this  locus  is  also  an  envelope,  its  equation  will  not  satisfy  the  differ 
ential  equation.  The  usual  case  is  illustrated  by  Fig.  2,  the  excep 
tional  case  by  Fig.  3. 

An  inspection  of  Fig.  2  will  show  why  the  equation  of  the  nodal  locus  (which, 
in  general,  is  not  an  envelope)  should  be  obtained  when  looking  for  the  equa 
tion  of  the  envelope.  In  this  figure  we  have  three  neighboring  curves,  which 


DIFFERENTIAL   EQUATIONS 


§33 


coincide  in  the  limit.  The  nodal  locus  is  indicated  by  the  broken  line.  Any 
point  on  it,  such  as  c,  is  the  limiting  position  of  the  point  of  intersection  of  the 
middle  curve  with  either  of  the  neighboring  ones,  i.e.  it  is  the  limiting  position  ot 
a  or  b.  But  while  in  the  case  of  the  envelope,  Fig.  i,  §  29,  a  and  b  approach 
coincidence  as  consecutive  points  on  the  middle  curve,  in  the  case  of  Fig.  2,  a 
and  b  approach  coincidence  in  an  entirely  different  way.  Consecutive  points 
are  points  which  approach  coincidence  by  moving  along  the  same  branch  of  a 
curve.  In  order  to  conclude  from  the  theorem  of  §  29  that  the  equation  of  the 
envelope  is  a  singular  solution  (§  30),  we  must  use  the  term  tangent  in  the  narrow 
sense  of  a  line  through  two  consecutive  points.  If  we  use  it  in  the  broader  sense 
of  a  line  through  any  two  coincident  points,  the  nodal  locus  may  be  said  to  be 
tangent  to  some  curve  of  the  family  at  each  of  its  points. 

A  special  case  of  a  double  point  is  a  cusp,  which  may  be  looked 
upon  as  the  limiting  case  of  a  double  point,  where  the  loop  has 
shrunk  up  to  the  point  and  the  two  branches  of  the  curve  have  be 
come  tangent.  The  equation  of  the  locus  of  the  cusps  of  the 
integral  curves,  known  as  the  cuspidal  locus,  found  when  the  equa 
tion  of  the  envelope  is  sought,  will  be  a  solution  only  in  case  this 
locus  is  also  an  envelope  (as  in  case  of  Fig.  4).  Otherwise,  it  is 
not  a  solution  (as  in  Fig.  5). 

In   the   case   of   a   cusp,  not  only  is  the 
number  of  integral  curves  through  that  point 
at  least  one  less  than  the  usual  number,  that 
is,  not  only  does  the  tr-discriminant  vanish  at 
this  point,  but  two  values  of/  are  equal  there, 
since  the  tangents  to 
the  two  branches  of 
the   curve  coincide, 
that   is,    at    such   a. 

[  c  point  the  /-discrinv 

FIG.  4  FIG.  5  inant    also  vanishes. 

Hence  the  equation  of  the  cuspidal  locus  must  also  appear  in  the 
/^-discriminant  relation. 


§33  SINGULAR   SOLUTIONS  71 

So  far,  these  extraneous  loci,  which  may  or  may  not  be  solutions, 
have  arisen  as  results  of  peculiarities  of  the  integral  curves.  Thus, 
if  the  integral  curves  are  known  to  have  no  double  points  or  cusps, 
it  is  clear  there  can  be  no  nodal  or  cuspidal  loci.  But  an  extraneous 
locus  may  arise,  irrespective  of  the  character  of  the  integral  curves. 
Wherever  two  distinct  integral  curves  are  tangent  to  each  other, 
while  the  number  of  curves  through  that  point  is  unaffected,  the 
number  of  distinct  values  of/  is  diminished.  Hence  the  /-discrimi 
nant  vanishes  at  that  point,  and  the  locus  of  such  points,  if  it  exists, 
will  be  given  by  the  /-discriminant  relation.  This  locus  is  known  as 
the  tac-locus,  and  its  equation  may  or  may  not  satisfy  the  differential 
equation.  Thus  in  the  case  of  the  family  of  circles  referred  to  in 
the  footnote,  page  61,  the  /-discriminant  relation  is  found  to  be 
/(/*  —  r2)  =  o  ;  here  y  =  ±  r  is  the  envelope,  while  y  =  o  is  the  tac- 
locus.  By  actual  trial,  it  is  found  that  y  —  o  does  not  satisfy  the 
differential  equation. 

Remark.  —  At  times,  as  the  parameter  approaches  a  certain  value, 
the  curves  of  the  family  approach  a  limiting  one,  usually  different  in 
shape  from  all  the  others.  Frequently  this  special  curve  of  the  family 
has  the  property  of  being  tangent  to  all  the  others  at  one  point.  Ex 
cepting  at  this  point  (through  which  there  is  an  infinite  number  of 
curves),  a  smaller  number  of  curves  than  usual  pass  through  every 
point  of  the  special  curve.  Hence  the  equation  of  the  latter  is  given 
by  both  the  /-  and  ^-discriminant  relations.  Moreover,  it  is  found 
that  the  factor  corresponding  to  this  special  solution  appears  once  in 
the  ^-discriminant  and  three  times  in  the  /-discriminant. 

Thus,  in  the  case  of  Ex.  4,  §  24,  the  integral  curves  are  a  family  of  cubica 
tangent  to  the  axis  of  y  at  the  origin  (see  Ex.  8,  §  20).  Their  equation  is  j2  = 
2x(x  —  <r)2.  For  c  =  oo ,  we  have  the  curve  x  =  o,  which  is  tangent  to  every 
other  one  at  the  origin.  The  c-  and  /-discriminants  are  respectively  xy2  and  x3. 
The  additional  factor  x1  appears  because  x  =  o  is  also  a  tac-locus,  it  being  a  par 
ticular  solution  corresponding  to  the  two  distinct  values  c  =  ±  oo . 


72  DIFFERENTIAL   EQUATIONS  §33 

Besides,  in  1888,  Mr.  J.  M.  Hill*  proved  that  the  factors  in  the 
^•-discriminant  corresponding  to  the  envelope,  nodal,  and  cuspidal 
locus  occur  once,  twice,  and  three  times  respectively,  while  those 
corresponding  to  the  envelope,  tac-locus  and  cuspidal  locus  in  the 
/-discriminant  occur  once,  twice,  and  once  respectively.  All  this  can 
be  put  in  tabular  form,  as  follows  : 


c-discriminant 

/-discriminant 

envelope 
particular  curve 
nodal  locus 

i 
i 

2 

envelope 
particular  curve 
tac-locus 

i 
3 

2 

cuspidal  locus 

3 

cuspidal  locus 

I 

In  case  any  locus  comes  under  two  heads,  the  factor  corresponding 
will  occur  the  number  of  times  it  should  for  each  of  the  heads  ;  thus, 
if  the  tac-locus  is  also  an  envelope,  that  factor  will  occur  once  in  the 
^•-discriminant  and  three  times  in  the  /-discriminant  ;  and  if  the 
cuspidal  locus  is  an  envelope,  it  will  occur  four  times  in  the  ^-dis 
criminant  and  twice  in  the  /-discriminant,  and  so  on. 

This  will  often  prove  a  check  on  the  work,  although  it  should  be 
only  relied  upon  as  a  check  and  not  as  the  only  clew  to  identify  the 
results  of  finding  the  discriminants.  The  process  of  finding  discrimi 
nants  is  frequently  beset  with  chances  to  introduce  or  to  drop  a 
factor,  so  that,  unless  great  care  is  taken,  the  number  of  times  a  factor 
is  found  to  occur  may  not  be  the  correct  one,  and  inferences  drawn 
from  it  will  be  false.  In  actual  practice  it  is  desirable  to  find  both  the 
/-  and  the  ^-discriminants,  and  then  test  their  various  factors,  equated 
to  zero,  to  see  if  they  satisfy  the  equation.  In  this  way,  if  a  factor 
has  been  lost  in  either  one  of  the  discriminants,  its  appearance  in  the 
other  will  keep  it  from  being  lost  as  a  solution,  while  a  blind  use  of  the 
table  would  cause  one  to  give  a  different  interpretation  to  the  result 

*  Proc.  Lond.  Math.  Society,  Vol.  XIX,  p.  561. 


§33  SINGULAR   SOLUTIONS  73 

In  certain  cases  there  can  be  no  doubt.  Thus,  if  the  degree  of 
the  equation  is  two  or  three,  the  use  of  the  formulae  mentioned  in 
§  31  will  give  all  the  factors  occurring  the  correct  number  of  times. 
Again,  in  case  the  integral  curves  are  straight  lines  (as  is  always  the 
case  when  the  equation  is  in  Clairaut's  form),  there  is  no  need  of 
looking  for  any  of  the  extraneous  loci. 

Again,  if  the  integral  curves  are  conies,  there  can  be  no  nodal  or 
cuspidal  loci. 

Examine  the  following  equations  for  singular  solutions  and  extra 
neous  loci  : 

fix.  1.   xf  —  (x  —  i)2  =  o. 

The  general  solution  is  readily  seen  to  be 


which  is  the  equation  of  a  family  of  nodal  cubics,  each  of  which 
is  tangent  to  the  axis  of  y  and  has  its  node  at  the  point  (3,  c)  . 

The  /-discriminant  relation  is  x(x—  i.)2  =  o,  while  the  ^-discrimi 
nant  relation  is  x(x  —  3)2  =  o. 

Here  x  —  o  is  common  to  the  two.  It  also  satisfies  the  equation. 
[For  the  line  x  =  o,  /  =  co  at  every  point.]  Hence  it  is  the  singular 
solution. 

.%•  —  i  =  o  occurs  in  the  /-discriminant  only.  It  is  the  tac-locus. 
[Notice  that  this  factor  occurs  twice.] 

x  —  3  =  o  occurs  in  the  ^-discriminant  only.  It  is  the  nodal  locus. 
[Notice  that  this  factor  occurs  twice.] 


Ex.  2.     8  (i  +/)3  =  27(*  +?)  (i  -/) 
The  general  solution  is 


74  DIFFERENTIAL   EQUATIONS  §33 

As  it  is  rather  awkward  to  substitute  the  coefficients  in  the  formula 
for  the  discriminant  given  in  §  31,  make  the  substitution 


The  equation  then  becomes 


and  the  solution  becomes  (77  +  cf  =  g2- 

Now  the  /-discriminant  relation  is  £2  =  o,  and  the  ^-discriminant 
relation  is  £4  =  o. 

£  =  o  is  common  to  both,  and  satisfies  the  equation.  Hence  £  =  o, 
or  x  +y  =  o,  is  a  singular  solution. 

It  is  also  a  cuspidal  locus,  as  may  be  seen  by  constructing  some 
of  the  semicubical  parabolas  (r)  +  c)s  =  g*.  (See  Fig.  4.)  Note  the 
number  of  times  that  these  factors  occur. 

Ex.  3.     4/2  =  9  x. 

The  general  solution  is 

fy+cfmaf. 

Here  the  /-discriminant  relation  is  x  =  o,  and  the  ^-discriminant 
relation  is  x3  =  o. 

It  is  obvious  that  x  =  o  does  not  satisfy  the  equation.  It  is  a  cuspi 
dal  locus. 

Ex.  4.  Examine  the  following  equations  for  singular  solutions  and 
extraneous  loci  : 


§24,  Ex.  3,4,  5.     §25,  Ex.  5,6.     §  26,  Ex.  2,  4.     §  27,  Ex.  2,  6. 
§  28,  Ex.  i,  2,  3,  5,  IT. 

Ex.  5.  The  family  of  circles  determined  by  Ex.  5,  §  26,  envelops 
a  curve  whose  equation  is  a  singular  solution  of  the  differential  equa 
tion.  Find  it. 


§  34  SINGULAR   SOLUTIONS  75 

34.  Summary.  —  We  have  seen  that  the  equation  of  the  singular 
solution  (or  of  the  envelope)  is  given  by  both  the  c-  and  /^-discrimi 
nant  relations  (§§  30,  32).  Moreover,  the  ^-discriminant  relation 
gives  rise  to  the  nodal  and  cuspidal  loci,  while  the  /-discriminant 
relation  gives  rise  to  the  cuspidal  and  tac-loci,  while  both  of  them, 
at  times,  give  rise  to  a  particular  solution  §  33.  For  the  number  of 
times  the  corresponding  factors  occur  in  each  discriminant,  see  re 
mark,  §  33.* 

Remark.  —  It  should  be  noted  that,  in  general,  a  differential  equation  has  no 
singular  solution.  For/(/,  x,y}  —  o  and^-  =  o  can  be  solved  for  y  and/ 
giving 


In  order  that  this  value  of  y  be  a  solution  we  must  have 


which  is  not  true,  in  general.  Darboux  proved  that,  in  general,  the  result  of 
eliminating/  from  the  above  two  equations  is  the  equation  of  the  cuspidal  locus. 
^Bulletin  des  Sciences  Mathematiqties,  1873,  p.  158.)  Picard  also  gives  a  proof 
of  this  in  his  Traite  d'  "Analyse,  Vol.  Ill,  p.  45.  See  also  Fine's  article  in  the 
American  Journal  of  Mathematics,  Vol.  XII  ;  Chrystal,  Nature,  1896;  Liebmann, 
Differ  entialgleichungen,  p.  95.  This  may  seem  at  first  sight  contrary  to  what  is 
to  be  expected  from  the  way  in  which  the  idea  of  a  singular  solution  was  intro 
duced.  (It  was  Lagrange  (1736-1813)  who  first  noted  that  the  equation  of  the 
envelope  of  the  family  of  integral  curves  is  a  solution.)  But  it  has  already  been 
noted  that  in  the  process  of  finding  the  equation  of  the  envelope,  extraneous  loci 
may  arise,  and  it  turns  out  that  these  usually  do  arise  to  the  exclusion  of  an 
envelope  (see  Picard,  Vol.  Ill,  p.  51).  Moreover,  all  this  was  based  on  the 
assumption  that  the  general  solution  of  the  equation  has  the  form  (i),  where  c 
enters  rationally.  While  this  is  true  in  a  very  large  class  of  equations,  it  is  never 
theless  only  a  special  case. 

*  The  theory  as  given  here  was  first  developed  by  Arthur  Cayley  (1821-95),  Messen 
ger  of  Mathematics,  Vol.  II  (1872),  p.  6,  Vol.  VI,  p.  23.  For  illustrative  examples  see 
].  W.  L.  Glaisher,  Messenger  of  Mathematics,  Vol.  XII,  p.  i. 


CHAPTER  VI 

TOTAL  DIFFERENTIAL   EQUATIONS* 

35.    Total  Differential  Equations.     A  differential  equation,  involv 
ing  three  or  more  variables,  of  the  form 

(i)  P(*,y,  z)dx+Q  (x,y,  z}dy  +  R  (x,y,  z)  dz  =  o 

is  called  a  total  differential  equation.     We  shall  consider  the  case 
when  its  solution  can  be  put  in  the  form 


The  differential  equation  arising  from  this  is 

xv  du   ,        du  ,        du  ,  

dx    '       by  J       ~dz   ' 

This  is  either  the  same  as  (i),  or  differs  from  it  by  a  factor 
p.(xt  y,  z)  ;  /.  e.  if  (i)  is  integrable,  there  must  be  an  integrating 
factor  for  it.  Then  a  function  p.(x9  y,  z)  exists,  such  that 

du          r,    du         ^.    du         n 

=  fJijT,      =  /JL  Q/,     =  /X/t . 

dx  dy  '    dz 

Hence 


BO 


smce 


*     ^~'         ^-^-^~^-> 

dz  dy  dy  dy  dz      dz  dy 


*  For  certain  reasons  it  seems  desirable  to  consider  this  class  of  equations  before 
going  to  the  study  of  differential  equations  of  higher  order  than  the  first.  If  desired, 
this  chapter  may  be  taken  up  after  Chapter  IX. 

76 


§35  TOTAL   DIFFERENTIAL   EQUATIONS  77 

dR  ^   -fy  dP,     pdiL  4    d*u  d*u 

P~^-  +  ^^T=  A*  T~  +  ^^    >  smce  5~^~==^~~^~> 

d.#  oje  02  02  02  cw      ox  02 

*"          * 


.ce 


w  07      ay  ox 


y.  must  satisfy  these  three  equations,  which  of  course  cannot  be 
expected  of  it,  unless  P9  Q,  R  satisfy  a  certain  condition  or  con 
ditions. 

If  we  multiply  these  equations  by  Pt  Q,  R  respectively  and  add, 
all  of  the  derivatives  of  p  disappear,  and  we  have  left 


after  dropping  the  common  factor  /x,  which  is  not  zero,  for  the  in 
troduction  of  zero  as  an  integrating  factor  gives  us  no  information. 

This  is  a  necessary  condition  among  the  coefficients  P,  Q,  R. 
Moreover,  we  shall  prove  that  it  is  the  only  condition  requisite  for 
the  existence  of  an  integral  of  (i),  in  other  words,  we  shall  prove 
that  this  condition  is  also  sufficient. 

Consider  any  one  of  the  variables,  say  2,  as  a  constant  temporarily. 
Then  equation  (i)  takes  the  form 

(4) 


*  Assuming  the  continuity  of  ^P,  ^Q,  n-R,  and  the  existence  and  continuity  of  theil 
derivatives. 

t  This  may  be  written  in  the  following  symbolic  determinant  form : 

P  Q  R 
o  o  o 
dx  dy  02 
P  Q  R 

which  is  very  easy  to  retain  in  mind. 


DIFFERENTIAL   EQUATIONS 


§35 


Still  considering  z  as  constant,  (4)  can  be  integrated,  but  now  the 
constant  of  integration  may  involve  z.     Let  the  solution  be 


(5) 


u  (x,y,  2)  = 


We  shall  show  that  if  the  condition  (3)  is  satisfied,  we  can  choose 
<f>(z)  so  that  (5)  will  be  the  solution  of  (i).  For,  differentiating 
(5)  with  respect  to  all  the  variables,  we  have 


(6) 


Since  (5)  is  a  solution  of  (4)  considering  z  as  constant,  we  have 
(  \  —  —    (  \  p    ^u  —     (  \  O 


du 
»   — 


where  /x  is,  in  fact,  an  integrating  factor  of  (4).     (See  §  5.) 
Comparing  (6)  with  (i)  multiplied  by  /x,,  we  have 

This  equation  can  be  solved  for  <f>  provided pR  reduces  to  a 

oz 

function  of  z  and  <£,  when  use  is  made  of  (5),  or,  what  is  the  same 


is  a  function  of  z  and  u.     Looking  upon 


thing,  provided  —  — 

dz 

—  —  fjiR  and  u  as  .  functions  of  x  and  y  only,  z  being  treated  as  a 
dz 
constant  or  parameter,  the  only  requirement  for  this  is  the  vanishing 

of  their  Jacobian  :  * 


du 

d*u. 

dR 
*"dx 

dR 

rjd/JL 

-*d^ 
~~RTy 

dx' 

du 

dx  dz 
d~u 

dy3 

dydz      ' 

1  By 

*  See  Note  I  in  the  Appendix. 


§35 


TOTAL  DIFFERENTIAL   EQUATIONS 


Making  use  of  (7),  this  may  be  written 

o      dP 

>  ^T" 

oz 


dR 

^  x 

ox          ox 


dQ  i   n9^        aR 

-IT  +  QZ-P^  —     x 

dz  oz          oy  dy 


or      fi."\  f (  -^-~  }+  Q  ^--~)  \-^R(P^- 

\dz        dyj          \dx       dzj_\ 

Assuming  that  (3)  is  satisfied,  this  becomes 


or 


dx 


dy 


Since  /x  is  an  integrating  factor  of  (4),  this  vanishes.  Hence  equa 
tion  (8)  can  be  solved  for  <£  ;  putting  this  in  (5),  we  have  the  solution 
of  our  equation  (i),  and  our  theorem  is  proved.* 

*  The  necessity  and  sufficiency  of  the  condition  (3)  can  be  proved  much  more 
briefly  as  follows  (but  this  method  does  not  suggest  a  general  way  of  solving  a  total 
differential  equation  when  the  condition  is  satisfied)  : 

The  equation  Pdx  +  Qdy  +  Rdz  =  o 

is  equivalent  to  the  two  partial  differential  equations, 
dz  =  __P      fo=_Q 

dx-      R'  dy      R' 

In  order  that  these  may  hold  simultaneously,  it  is  necessary  and  sufficient  that 


dy 


dx 


Remembering  that/3,  Q,  R  are  functions  of  x,y,  z,  this  equation  becomes 
K(^+^te\_p(^.^dz\_ 

\dy^  dzdy)      \dy  ~^  dz  dy)~ 


Since  —-=  —  —  ,    —  =  —  ^-,  this  reduces  at  once  to  the  form  (3)  above. 
QX         R     oy         R 


80  DIFFERENTIAL   EQUATIONS  §36 

36.  Method  of  Solution. — The  above  proof  not  only  establishes 
the  sufficiency  of  the  condition,  it  also  suggests  the  following  method 
for  solving  a  total  differential  equation  in  three  variables  which  satis 
fies  this  condition  : 

Integrate  the  equation  considering  one  of  the  variables  *  as  a  con 
stant.  Instead  of  a  constant  of  integration,  introduce  an  undetermined 
function  of  this  variable.  Re  differentiate,  this  time  with  respect  to 
all  the  variables.  Comparing  this  with  the  original  differential  equa 
tion,  a  new  differential  equation  will  arise,  involving  only  the  undeter 
mined  function  and  that  variable  of  which  it  is  a  function.  From 
this  the  function  can  be  determined,  involving  an  arbitrary  constant. 
And  thus  the  complete  solution  is  found. 

Remark.  —  Since  an  equation  which  is  integrable  differs  only  by  a 
factor  from  an  exact  differential  equation,  if  we  can  obtain  such  a 
factor  by  inspection  or  otherwise,  we  can  integrate  at  once. 

Apply  test  for  integrability  and  integrate  the  following: 


Ex.  l.  fdx  +  zdy  —y  dz  =  o. 

=/(-  i  - 1)  -M(o  -  o)  — Xo  -  2 y)  =  -  2/  -|-  2/  =  o. 


z    -y 
_d_     d   _d 

dx    dy    dz 

/    z    -y 


While  in  this  simple  case  there  is  very  little  choice  in  selecting  the 
variable  to  be  constant,  there  is,  perhaps,  a  little  advantage  in  let 
ting  y  be  so  chosen.  Then  we  have 

y*dx—ydz  =  o,   whence yx  —  z  —  <{>(y). 
Differentiating,  y  dx  -}-  x  dy  —  dz  =  d<f>, 

or  y2  dx  +  xy  dy  —  y  dz  =yd<f>. 

*  We  usually  choose  that  variable  a  constant  which  will  have  the  effect  of  simplify* 
ing  the  resulting  equation  in  the  other  two  variables  as  much  as  possible. 


§37  TOTAL  DIFFERENTIAL   EQUATIONS  8  1 

Comparing  this  with  the  differential  equation,  we  have 

(xy-z)dy=yd$, 
or  $dy 

/.  <£  =  cy. 
Hence  the  general  solution  is  yx  —  z  -f-  cy  =  o. 

By  inspection,  it  is  readily  noticeable  that  —  is  an  integrating 
factor.     This  puts  the  equation  in  the  form         -^ 

zdy  —  ydz 
dx  +     ^    /     =  o. 

Its  solution  is  of  course  x  —  -  -f  c  =  o. 


Ex.2,   zy  dx  —  zxdy—  y*dz  =  o. 


Ex.3,   xdx  -\-ydy  —  V#2  —  x2  —  y" 

Ex.4,    (x?  —  y2  —  z*)dx-\-  2xydy-\-2  xzdz  =  o. 


37.  Homogeneous  Equations.  —  If  P,  Q,  and  R  are  homogeneous 
and  of  the  same  degree,  the  variables  may  be  separated  just  as  in  the 
corresponding  case  for  two  variables  (§  10).  Here  we  transform  any 
two  of  the  variables,  say  x  and  y,  by  x  =  uz,  y  =  vz.  Then  dx  =  zdu 
+  udz,  dy  =  zdv  +  vdz,  and  the  differential  equation  becomes 

z(Pl  du  +  <2i  dv)  +  (*/>!  +  »<?i  +  R\)&  =  °*  or 


,s  iu  z  = 


z 


where       Pl  =  P(u,  v,  i),  ft  =  Q(u,  v,  i),  R,  =  R(u,  v,  i). 

*  If  uP1  +  vQi  +  Xi  =  o,  this  equation  reduces  at  once  to  one  in  the  two  variables 
u  and  v. 


82  DIFFERENTIAL   EQUATIONS  §37 

Now,  if  the  original  equation  satisfies  the  condition  for  integra 
lity,  this  equation  will  also.  Moreover,  it  is  exact*  and  can  be 
integrated  as  it  stands  (by  method  of  §  8). 

Ex.  1.    (/  +yz)dx  +  (xz  +  z*)dy  +  (/  -  xy]dz  =  o. 

(Let  the  student  apply  the  test  for  integrability.) 

Putting  x  =  uz,  y  =  vz,  dx  =  u  dz  +  z  du,  dy  =  vdz-\-  z  dv,  and  the 
equation  becomes 


dz_      (v*  +  v)du-\-  (u+  i)dv  _ 
z  uv^  +  uv  -\-v 


Since   uv*  +  uv  +  v  +  v2  =  (v*  +  v')(u+  1),    the    equation   may  be 

written 

dz       du  dv 

I        ; 
z       u  +  i 


whence 


or  ** — : — '-  =  c. 


Ex.  2.    (f  +yz  +  &)dx  +  (z2  +  zx  +  x*)dy  +  (x2  +  xy  +f)dz  =  o. 
Ex.  3.    (x°y  -/  -y*z)dx  +  (#/  -  x>z  -  x*)dy  +  (xy2  +  £y)di  =  o. 


Putting 


«ft4-.*-fA 


equation  (i)  takes  the  form  Pdu  +      ^v+  R  dz  —  o. 

Since  T3  and  ^  are  free  of  z,  and  ^  is  free  of  u  and  z>,  the  condition  for  integrability 
reduces  to 


A"  is  -,    hence  we  must  have  ^5  — ^=o,  which  means  that  Pdu+Qdv  is  an 

du      dv 
exact  differential  (§  7),  and  therefore  (i)  is  also. 


§  38  TOTAL  DIFFERENTIAL   EQUATIONS  83 

38.   Equations  involving  more  than  Three  Variables.  —  Consider 
the  equation 
(i) 


If  this  is  integrable,  it  will  remain  so  when  we  let  any  of  the  varia 
bles  be  a  constant.  Letting  x,  y,  z,  t  be  constants  successively,  the 
conditions  are 


dS     5Q\  5P     d_ 

~~  ~-  ' 


But  these  conditions  are  not  all  independent.  If  we  multiply  (2), 
(3),  (4)  by  P,  Q,  R  respectively  and  add,  we  get  (5)  multiplied 
by  S  ;  which  shows  that  only  the  first  three  are  independent.! 

*  Perfectly  generally,  if  the  equation  contains  n  variables,  we  obtain  as  many  con 
ditions  as  the  number  of  ways  in  which  we  can  pick  out  three  variables  from  *  ;  that 

is,  the  number  of  conditions  is  "("""^-(JLngl. 

3! 
f  In  general,  the  number  of  independent  conditions  in  the  case  of  n  variables  is 

(n—i)(M  —  2)  ^  wnjcn  js  tne  number  of  times  two  objects  can  be  chosen  out  of  n  —  i. 

2 

For  only  those  conditions  will  be  independent  which  involve  derivatives  with  respect 
to  some  one  chosen  variable,  since  any  condition  not  involving  such  can  be  obtained 
by  combining  linearly  those  that  do,  as  was  done  in  the  case  above.  Now  each  of  the 
conditions  involves  derivatives  with  respect  to  three  variables.  Hence  the  derivatives 
with  respect  to  any  one  variable  may  appear  in  a  condition  along  with  those  with 
respect  to  any  two  of  the  remaining  n  —  i  variables. 


84  DIFFERENTIAL   EQUATIONS  §39 

These  conditions  can  also  be  shown  to  be  sufficient.  When  they 
hold,  the  integral  is  found  as  in  the  case  of  three  variables,  by  in 
tegrating,  considering  all  but  two  of  the  variables  constant.  Then 
the  constant  of  integration  is  written  as  a  function  <f>  of  those  vari 
ables  temporarily  considered  constant.  Redifferentiating  with  respect 
to  all  the  variables,  and  comparing  with  the  given  equation,  the  two 
variables  originally  treated  differently  from  the  rest  will  disappear, 
and  the  function  will  enter  in  a  new  differential  equation  which  is 
integrable  and  involves  the  remaining  n  —  2  variables  and  <£,  that  is 
n  —  i  variables  in  all.  Either  this  can  be  integrated  at  once,  or  the 
process  may  be  repeated  as  often  as  necessary.  The  following  ex 
ample  will  illustrate. 

Ex.     z(y  +  z)dx  +  z(t  -  x)dy  +y(x  -  i}dz  +y(y  +  z)dt=  o. 
Let  y  and  z  be  constants  temporarily.     Integrating,  we  have 

xz+yt=<f>(y,z). 
Differentiating  and  comparing  with  the  original  equation,  we  have 

(ty  +  zx)(dy  +  dz)  =  (y 
or  $ 


We  now  have  an  equation  in  the  three  variables  y,  z,  <£.     This  can 
be  solved  by  the  general  method  (§  36).     But  an  obvious  integrating 

factor  is  (         .  ,  •        Introducing  this,  we  have 


Hence  the  general  solution  is 


39.    Equations  which  do  not  satisfy  the  Condition  for  Integrability. 
If  Pdx  -f  Qdy  +  Rdz=o  does  not  satisfy  the  condition  for  Integra- 
bility,  it  is  impossible  to  find  its  general  solution  in  the  form 


§39  TOTAL   DIFFERENTIAL   EQUATIONS  85 

But  as  the  equation  is  one  in  three  variables,  we  should  expect  to 
find  an  indefinite  number  of  solutions.  As  a  matter  of  fact,  if  we 
assume  any  relation  we  please,  say  $(x,  y,  z)=o,  this  will  deter 
mine  any  one  of  the  variables,  say  z,  in  terms  of  the  other  two. 
Substituting  for  this  variable  in  the  original  equation,  we  obtain  a 
new  differential  equation  in  two  variables,  which  can  usually  be 
solved.  We  see  then  that  the  general  solution  of  a  so-called  non- 
integrable  total  differential  equation  consists  of  an  arbitrarily  chosen 
relation  among  the  variables  and  a  second  relation  involving  an 
arbitrary  constant.  The  latter  depends  upon  the  choice  of  the 
former,  and  cannot  be  determined  until  the  choice  has  been  made. 

Remark.  —  Since  the  solution  of  the  integrable  equation  is  a  single 
relation  among  the  three  variables,  we  may  assume  any  second  one 
consistent  with  it.  So  that  in  this  case  also  we  may  say  that  the 
solution  consists  of  an  arbitrarily  chosen  relation  and  a  second  one 
involving  an  arbitrary  constant.  But  here  the  latter  is  fixed  by  the 
differential  equation,  and  is  entirely  independent  of  the  choice  of  the 
former. 

Ex.  y  dx  -f  x  dy  —  (x  +j+  z)  dz  =  6. 


This,  it  is  readily  seen,  does  not  satisfy  the  condition  for  integra- 
bility.     If  we  assume  x  -f  y  +  z  =  o,  our  equation  becomes 

y  dx  -f-  x  dy  =  o,  whose  solution  is  xy  =  c. 
Hence  a  solution  is  J 


If  we  assume  x  -\-y=  o,  our  equation  becomes  y  dx  -f-  x  dy  —  z  dz  =  o, 
whose  solution  is  2  xy  —  22  —  c.    Hence  another  solution  is 


{2  xy  —  z2=c. 


86  DIFFERENTIAL   EQUATIONS  §40 

40.   Geometrical  Interpretati  i.  —  To  say  that  the  equation 

$  $x+  Qdy    -Rdz^v 
satisfies  the  condition  for  int  ^ability  is  to  say  that  a  family  of  surfaces 


exists  such  that  at  each  point  (XQ,  yQ,  z0)  in  space  there  passes  one* 
of  these  surfaces 


and  the  tangent  plane  at  any  point  (x,y,  z)  of  this  surface  is 
P(xty,  z)  (X-  x)  +  Q(x,y,  z)(Y-y)  +  R  (x,y,  z)  (Z-  z)=o 

In  other  words,  the  differential  equation  defines  the  plane 
P(X—  x)  +  Q  (  Y—y)  +  R  (Z—z)  =  o  at  each  point  in  space.  The 
problem  of  integration  amounts  to  determining  a  family  of  surfaces! 
such  that  the  surface  which  passes  through  any  point  is  tangent  to 
the  plane  corresponding  to  that  point.  An  interesting  consequence 
of  this  is  brought  out  in  §  66. 

When  the  equation  is  not  integrable,  the  assumption  of  a  second  re- 

di[/  d{f/  d\l/ 

lation,  \J/  (x,  y,  z)  =  o,  which  carries  with  it  ^—  dx  -f-  -*rdy  +  ~Q~^Z  —  °> 

determines,  with  the  original  equation,  a  line  at  each  point  on  the 
assumed  surface  \j/(x,  y,  z)  =  o,  viz. 

P(X-x)  +Q(Y-y) 


The  problem  of  integration  then  amounts  to  determining  a  family 
of  curves  such  that  that  curve  which  passes  through  any  point  is  tan 
gent  to  the  line  corresponding  to  that  point.  Since  one  of  the  two 

*  It  is  presupposed  here  that  0  is  a  rational  function  of  x,y,  z.     Otherwise  the  state 
ment  in  the  text  must  be  restricted  to  regions  in  which  0  is  single-valued.     (See  §  70.) 
f  These  will  be  referred  to  as  integral  surfaces. 


§  4i  TOTAL  DIFFERENTIAL   EQUATIONS  87 

equations  of  this  family  of  curves  is  th  'issumed  relation  if/(x,y,  z)  =  o, 
the  problem  really  amounts  to  fmdint  that  famil  -f  curves  on  any 
arbitrarily  chosen  surface  whose  tange;  at  any  point  of  the  surface 
lies  in  the  plane  determined  by  the  differ  "^tial  equation  at  that  point. 
Thus,  in  the  case  of  Ex.,  §  39,  we  have  c  the  plane  x-\-y  +  z=o 
the  curves  cut  out  upon  it  by  the  fanL  y  of  cylinders  xy  =  c  ; 
while  on  the  plane  x+y  =  o  we  have  the  curves  cut  out  by  the 
hyperbolic  paraboloids 


41.   Summary.  —  If  the  total  differential  equation 
Pdx 


satisfies  the  condition  for  integrability  (§  35),  an  integrating  factor 
exists.  If  that  can  be  found  by  inspection,  introduce  it,  and  inte 
grate  at  once. 

If  the  integrating  factor  cannot  be  found  by  inspection,  the  gen 
eral  method  of  §  36  may  be  employed. 

If  P,  Q,  R  are  homogeneous  and  of  the  same  degree,  the  method 
of  §  37  will  sometimes  prove  simpler  than  the  general  method. 

If  the  condition  for  integrability  is  not  satisfied,  solutions  may  be 
found  by  the  method  of  §  39. 

Total  differential  equations  involving  more  than  three  variables 
may  be  treated  by  the  method  of  §  38,  unless  an  integrating  factor  is 
obvious  by  inspection.  In  this  case  introduce  the  factor  and  inte 
grate  at  once. 

Apply  the  test  for  integrability,  and  solve  the  following  : 

Ex.l.   (y  +  *)dx  +  (*  +  x)Jy+(x+y)dt  =  <y, 

Ex.2.    (z+i)(xdx+ydy)-(x*+yl)dz  =  o. 

Ex.3. 

Ex.4. 


*  All  this  applies  to  integrable  equations,  except  that  in  case  the  arbitrarily  chosen 
surface  is  an  integral  surface,  every  curve  on  it  is  an  integral  curve. 


88  DIFFERENTIAL   EQUATIONS  §41 

Ex.    5.  (y  +  z)dx  +  dy  +  dz  =  o. 

Ex.    6.  2  x  dx  +  dy-\-(2  x^z  -\-  2  yz  -{-  2  z*  -\-  \]dz=  o. 

Ex.    7.  (2  x+f-\-2xz)dx-\-2  xydy  +  x*  dz—dt=v. 

Ex.    8.  zxdy—  yzdx  +  x2dz  =  o. 

Ex.    9.  x(y—i)(z—  \)dx+y(z-  i)(x-  \}dy 

+  z(x—  i)(y—  i)dz  =  o. 

Ex.  10.  (y  —  z}dx  +  2  (x  +  3  y  —  z)dy  —  2  (x  +  2  j')^/s  =  o. 

Ex.  11.  /(j  +  2)^r-h/t(_y  +  z+  i)</F  +  /</3—  (_y  +  s)^/=o. 

Ex.  12.  2(7  +  z)dx  +  z(t—  x}dy  +y(x  —  t)dz  +y(y  +  z)dt=  o. 


CHAPTER   VII 

LINEAR  DIFFERENTIAL  EQUATIONS  WITH  CONSTANT 
COEFFICIENTS 

42.  General  Linear  Differential  Equation.  —  A  linear  differential 
equation  is  one  which  is  of  the  first  degree  in  the  dependent  variable 
and  all  of  its  derivatives.  Its  general  type  is 


y     -i  Y        i  Y        \      ^Y 

°~7^"^"       *V    n    l"*"^^    n    2~*  -----  r  ^-n-l-i- 

dxn          dxn~  dxn~  dx 


where  Jf0,  J^,  Jf2,  •••>  XM  X  are  functions  of  x  or  constants.     If  we 

write  -*•  —  Z>y,  —4  =  Z)2 
dx  dx* 

lowing  convenient  form, 


write  -*•  —  Z>y,  —4  =  Z)2y,  ••  •,  —  2  =  Z^y,  we  can  write  (i  )  in  the  fol- 
dx  dx*  dxn 


or  F(D)y  = 


where  ^(Z>)  is    the  polynomial  ^o^1  +  X1£>n~1  +  •-  +  Xn  which 
represents  symbolically  the  differential  operator 

X^n  +  X*  71T1  +  '•'  +  X^T  +  x*- 
dxn  dxn~  dx 

Two  properties  of  linear  differential  equations  which  are  of  service 
in  their  solution  deserve  especial  mention  here  : 

i°  Suppose  X=  o.  In  this  case  the  equation  is  said  to  be  a  homo 
geneous  linear  differential  equation,  since  all  of  its  terms  are  of  the 
first  degree  in  y  and  its  derivatives.  (When  not  homogeneous,  the 


90  DIFFERENTIAL   EQUATIONS  §42 

equation  is  said  to  be  a  complete  linear  differential  equation.)  If 
y  =yi  satisfies  the  equation,  so  will  y  —  clylj  where  ^  is  a  constant. 
For,  since  -0*  fa  ^i)  =  rx /?«%  F(D)(c^)  =  c^F(D)y^  But,  by 
hypothesis,  F(D)y\  =  o,  hence  F(D)  (^i  j^i)  =  o. 

Moreover,  if  y  =  y2  is  also  a  solution,  y  =  cly}  +  c2y2  will  be  a 
solution.  For,  since  the  derivative  of  the  sum  is  the  sum  of  the 
derivatives,  i.e.  D*  (yl  +  y2)  =  Dky1  +  LPy^  we  have 


Similarly,  if  we  know  r  particular  integrals,  ylt  y2)  •••,  yr 


will  be  a  solution.  Since  the  general  solution  of  a  differential  equa 
tion  of  the  nth  order  is  a  solution  which  involves  n  independent  arbi 
trary  constants,  we  have  the  property  : 

A.  If  y\,  y-t,  "',yn  °^re  n  linearly  independent*  particular  integrals 
of  a  homogeneous  linear  differential  equation  of  the  nth  order,  the 
function  c±y±  +  c%y2  -{-  •"  +  cnynis  its  general  integral. 


If  the  particular  integrals  are  not  linearly  independent,  the  solution  found 
above  will  not  be  the  general  solution.  Thus,  suppose  there  exists  the  relation 
a\y\  +  ^2^2  +  •••  +  anyn  =  o,  where  all  the  #'s  are  not  zero.  If  an  is  different 

from    zero,  yn  =  —  —  y\  —  —  y<2.  —  •••  —  ^-^  yn  _  i,  and  the  integral  becomes 
an          an  an 


where  only  n  —  I  independent  constants  are  involved. 

*  n  functions  ylt  y%,  •••.j'n  of  a  variable  are  said  to  be  linearly  independent  if  it  is  impos 
sible  to  find  n  constants  alt  a>2,  •••,  an'such  that  a1yl  +  a2^2  +  '"  +  anyn  shall  vanish  for 
all  values  of  the  variables.  Thus  yl  =  2  x  —  *2,  j2  =  x  +  *2,  y$  =  x  are  evidently  not 
linearly  independent,  since  y\-{-y^  —  3^3  =  0;  i.e.  it  equals  zero  for  all  values  of  x,  or, 
as  it  is  usually  expressed,  y1  +y2  —  3J8  vanishes  identically. 


§43  LINEAR,   WITH   CONSTANT   COEFFICIENTS  91 

Remark.  —  Attention  should  be  called  to  the  fact  that  it  makes  no 
difference  how  these  particular  integrals  are  gotten.  We  shall  see 
that  in  a  most  commonly  occurring  class  of  equations,  these  will  be 
found  by  purely  algebraic  means  ;  in  other  cases,  some  of  them  can 
be  gotten  by  inspection. 

For  convenience  of  language,  the  integral"  of  (i),  when  the  right- 
hand  member  is  made  zero  temporarily,  is  spoken  of  as  the  comple 
mentary  function. 

2°  If  Y=  <:,  yl  +  c^  H  -----  V  cnyn  is  the  complementary  function 
of  (i),  and  if  we  know  (no  matter  by  what  means)  a  particular  in 
tegral,  (7,  then  F-f  U  is  the  general  integral  of  (i).  For,  since  the 
equation  is  linear, 

f(2))(Y  +  U}  =f(D)  Y+f(D)  U=o+X=  X. 

Hence  the  property  : 

B.  The  general  integral  of  a  complete  linear  differential  equation 
is  the  sum  of  its  complementary  function  and  any  particular  integral. 

43.  Linear  Differential  Equations  with  Constant  Coefficients.* 
Complementary  Function.  —  Given  the  equation 


or  (k^D*  +  k^D*^  +  V71-2  +  •  •  •  +  kn_^D  +  k^y  =  X, 

or  f(P)y=X, 

where  k^  k±,  •••,  kn  are  constants. 

First,  suppose  X  =  o.     Then 
(2)  /(^)^  =  o. 

*  The  method  given  here  is  due  to  Leonhard  Euler  (1707-1783).  For  a  presenta 
tion  of  Cauchy's  method  see  T.  Craig,  A  Treatise  on  Linear  Differential  Equations, 
Vol.  I,  Ch.  II;  or  C.  Hermite,  "Equations  Differentielles  Lineaires,"  in  Bulletin  des 
Sciences  Mathematiques,  1879. 


Q2  DIFFERENTIAL   EQUATIONS  §43 

Putting;/  =  emx,  we  have,  Dy  —  memx,  •••,  Dry  =  nfe™  ; 
hence, 


For  e™*  to  be  an  integral  of  (2),  m  must  satisfy  the  equation 
(3)  /(*)-o, 


Each  value  of  .m  satisfying  (3)  gives  an  integral  of  (2).  If  these 
are  all  distinct  (say  m1}  m2,  m3)  •••,  mn),emix,em2x,'",  emnx  will  be  linearly 
independent,  and  making  use  of  A,  §  42,  <r1<?mia!  -f  c^emf  +  •••  -f  r^n* 
will  be  the  general  integral  of  (2),  and  the  complementary  function 

Of(l). 

Remark.  —  Equation  (3),  which  is  so  readily  obtained  from  equa 
tion  (2),  is  usually  referred  to  as  the  auxiliary  equation.* 

Ex.l.    ^-3^  +  2J,  =  o. 
ax*        ax 

The  auxiliary  equation  is  m*  —  3  m  -+-  2  =  o.  Its  roots  are  i,  2. 
Hence  the  general  solution  is 


dx*        dx 

Here  m?  —  6  m  +  25  =  o,  whence  m  =  3  ±  4  z',  where  /=  V—  i. 
.-.  _y  =  <r1<?(3+4t)z  +  ^2<?(3~4t)l,    or  _y  = 

T^,           d*y      dy 
Ex.  3.    --* ^-  =  o. 


Ex.4.    (7>3-2J92 

*  Cauchy  calls  this  the  characteristic  equation. 


§  44  LINEAR,  WITH  CONSTANT  COEFFICIENTS  93 

44.  Roots  of  Auxiliary  Equation  Repeated.  —  If  any  roots  of  the 
auxiliary  equation  are  repeated,  the  method  of  §  43  does  not  give 
us  n  linearly  independent  integrals,  and  consequently  it  does  not 
give  us  the  general  solution.  In  this  case  we  make  the  more  general 
substitution  y  =  emx$(x)y  where  <$>(x)  is  a  function  of  x  entirely  at 
our  disposal.  Then 


2! 


whence 


If  ^!  is  an  r-fold  root  off(m)  =  o,  we  have  §  (31), 

/Oi)  =  o,  /'(///O  -  o,  .-,  /<-«(«)  =  o. 

In  this  case  f(D)  y  will  vanish  if  y  —  emx(f>  (x)  provided  Dr<}>  =  o, 
whence  all  the  higher  derivatives  of  <£  are  also  zero  ;  i.e.  pro 
vided  <f>  =  cv  xr~l  +  r^'"2  +  •  '  '  +  cr—\x  -f  ^«  where  r1}  ^2»  "*>  ^r  are 
any  constants  whatever.  Hence,  we  see  that  if  ml  corresponds  to  an 


94  DIFFERENTIAL   EQUATIONS  §45 

r-fold  root  of  the  auxiliary  eqjtation,  not  only  is  em^  an  integral  of  the 
equation,  but  so  also  are  xem*x,  x2emf,  •••,  xr~1emix,  i.e.  corresponding  to 
an  r-fold  root  we  have  r  linearly  independent  integrals.  So  that 
whether  the  roots  of  the  auxiliary  equation  are  repeated  or  not,  the 
n  linearly  independent  integrals  necessary  for  obtaining  the  com 
plementary  function  (A,  §  42)  are  always  supplied  by  the  auxiliary 
equation. 

Ex.1.    (4^-3^+1)^  =  0. 

The  roots  of  4  ms  —  3  m  +  i  =  o  are  i,  ^,  —  i.     Hence  the  gen 
eral  solution  is  y  =  e^x  (^  -f-  c^pc)  -\-  cze~x. 


Ex.2. 
Ex.3. 
Ex.4.  jy- 


45.  Roots  of  the  Auxiliary  Equation  Complex.  If  the  coefficients 
of  the  differential  equation  are  real,  while  some  or  all  of  the  roots  of 
auxiliary  equation  are  not,  we  can,  by  a  proper  arrangement  of  the 
terms  in  the  complementary  function,  have  the  latter  involve  only 
real  terms.  Thus,  if  the  auxiliary  equation  has  a  root  a,  -f  if},  it  will 
also  have  «  —  ij3  as  a  root,  since  its  coefficients  are  real.  Two  terms 
of  the  complementary  function  will  then  be 


or 


Now  e*Px  =  cos  fix  +  i  sin  fix,  and  e~{P*  =  cos  ftx  —  /'sin  fix. 
Hence  our  pair  of  terms  may  be  written 

£"*f  +      cos        +  it  —  £    sin    x. 


§45  LINEAR,   WITH   CONSTANT   COEFFICIENTS  95 

Putting  ti  =  A~~  iB,  c*  =  A  +  tB  ,        this  becomes 

2  2 

4°*(A  cos  fix  +  B  sin  (3x), 

where  A  and  B  are  the  two  arbitrary  constants. 

Another  form  in  which  this  may  be  written  is  ae**  sin  (fix  +  &)* 
or  at**  cos  (/£v  +  b),  where  a  and  b  are  the  arbitrary  constants. 
For  interpreting  the  solutions  in  physical  problems,  the  latter  forms 
are  sometimes  preferable. 

It  is  obvious,  that  in  case  a  pair  of  such  roots  is  repeated,  the 
corresponding  part  of  the  complementary  function  is 

eax  (Ai  cos  (3x  +  B±  sin  fix}  +  xeax(Az  cos  fix  +  Bz  sin  px) 
or       eax  l(A1  +  A2  x)  cos  (3x  +  (B±  +  B2  x)  sin  /£*]. 

And  perfectly  generally,  in  case  such  a  pair  occur  as  r-fold  roots,  the 
corresponding  part  of  the  complementary  function  is 


Ex.  1.     In  the  case  of  Ex.  2,  §  43, 
«  =  3,  /?  =  4,  so  that  the  solution  may  also  be  written 

y  =  <p*(A  cos  4  x  +  B  sin  4  x) 
or  y  —  a  <?*  cos  (4  x  +  b)  . 

Ex.  2.     (IP  +  2  Z>2  +  1)7  =  o. 
Ex.  3.      jy  -  /?  +  Z>     =  o. 


*For,  A  cos  px  +  B  sin  /3*   may  be    written   -VA^  +  B2  (     /          ^  cos 
.     Since   the   sum    of   the   squares   of  ;/==  and 


equals  unity,  these  may  be   taken    as    the   sine   and   cosine   of  some   angle,   say  b. 
Putting  V^2  _|_  /?a  =  a>  our   expression   becomes   a  (sin  ^  cos  j3^  +  cos  b  sin  /3^)   or 


96  DIFFERENTIAL   EQUATIONS  §46 

Remark.  —  For  the  purpose  of  interpreting  the  solution  of  certain  problems  in 
Physics  it  is  desirable  at  times  to  introduce  hyperbolic  functions  in  place  of  the 
exponentials  in  case  a  pair  of  the  roots  of  the  auxiliary  equation  are  real  and 
equal  to  within  the  sign.  Proceeding  as  before,  we  make  use  of  the  formulae 
ex  =  cosh  x  4-  sinh  x,  e~x  =  cosh  x  —  sinh  x. 

If  +  m  and  —  m  are  a  pair  of  roots  of  the  auxiliary  equation,  the  correspond 
ing  terms  of  the  complementary  function  are 


=  (/i  +  ^2)  cosh  mx  -j-  (^i  —  c  2)  sinh  mx 

—  A  cosh  mx  +  B  sinh  mx. 
Using  the  addition  theorem  for  the  hyperbolic  functions,  this  may  also  be  written 

y  =  a  cosh  (inx  +  ^)> 

or  y  =  a  sinh  (nix  -f  £)» 

where  a  and  b  are  arbitrary  constants. 

46.   Properties  of  the  Symbolic  Operator  (D  —  a).  — 

i°    (D-  a)y  means  ^  -  ay.     Similarly  (D  -  fi)y  means  ^  -  fiy. 
ax  ax 

Hence  \_(D  —  a)  -f  (D  —  /?)]  y  means    2  ^-  —  (a  +  ff)y,   which  may 

be  written  symbolically  [2  D—  («  +  j8)].y.  That  is,  the  result  of 
operating  on  7  with  (/?  —  a)  and  (D  —  ff)  separately  and  then  taking 
the  sum,  is  the  same  as  operating  on  y  with  [2  D  —  (a  +  /?)].  Hence 
we  see  that  the  operation  resulting  from  taking  the  sum  of  the  results 
of  two  operations  of  the  type  here  considered  can  be  gotten  symbolically 
by  taking  the  sum  of  their  symbolic  representatives.  Thus  we  can  write 


Evidently  this  rule  applies  to  the  sum  of  any  number  of  such  oper 
ators,  and  also  to  the  difference  between  any  two  of  them. 

2°    (D  —  (3)(-D  —  a)y    means    (-,  ---  0V^  —  «?],    which    is 

\  dx         )  \ax  J 

-~  —  («  -f-  /?)  -J  +  a/3y.     That  is,  the  result  of  operating  on  y  with 


§47  LINEAR,    WITH   CONSTANT  COEFFICIENTS  97 

(J}—a)  first,  and  then  with  (D  —  /3)  on  the  result,  is  the  same  as 
operating  on  y  with  [Z>2  —  (a  +  f$)D  -\-  a/3].  Hence  we  see  that  the 
operation  resulting  from  the  successive  performance  of  two  operations 
of  the  type  here  considered,  can  be  gotten  symbolically  by  taking  the 
product  of  their  symbolic  representatives.  Thus  we  can  write 


Moreover,  owing  to  the  symmetry  of  a  and  /3  in  the  right-hand 
member,  we  see  that  the  order  of  the  operators  on  the  left  is  not 
essential,  or,  as  it  is  usually  put,  two  operations  of  the  type  here  con 
sidered  are  commutative. 

Obviously  all  this  applies  to  any  number  of  operations  of  the  type 
here  considered. 

All  the  results  of  this  paragraph  can  be  incorporated  in  the  follow 
ing  : 

The  symbolic  representatives  of  operations  of  the  type  here  consid 
ered  behave  like  algebraic  quantities  for  the  processes  of  addition,  sub 
traction,  and  multiplication. 

Remark,  —  Since  any  polynomial  in  D  with  constant  coefficients  is  a 
product  of  linear  factors,  this  theorem  applies  also  to  operators 
whose  symbolic  representatives  are  polynomials  in  D  with  constant 
coefficients. 

Evidently  if  the  roots  of  the  auxiliary  equation  of  (i)  are  mit  m>2, 
•••,  mn  (whether  these  are  all  distinct  or  not),  we  may  write  (i)  in 

the  form 

mj)~.(D-  mn)y  =  X. 


47.  Particular  Integral.  —  A  perfectly  general  method  for  obtain 
ing  the  particular  integral  of  a  complete  linear  differential  equation 
with  constant  coefficients  (and,  for  that  matter,  another  method  for 
obtaining  the  complementary  function,  as  well)  results  from  the  fol 
lowing  considerations  : 


98  DIFFERENTIAL   EQUATIONS  §47 

In  the  following  discussion  we  shall  suppose  the  equation  divided 
through  by  kQ)  and  to  simplify  matters  and  yet  bring  out  the  method, 
we  shall  use  an  equation  of  the  third  order.  Let  us  start,  then,  with 
the  equation 

f(D)y  =  (D-  mj  (D  -  mj>  (D  -  m^y  =  X. 

To  find  the  integral  of  this  equation  is  to  find  y,  a  function  of  x, 
such  that  when  operated  on  by  f(D)  it  will  give  X. 

Let  (D—  m^(D  —  ms)y  =  u,  where  u  is  a  new  function.     Then 

(D  -  mju  =  X,or—-  mm  =  X. 

This  is  a  linear  equation  of  the  first  order,  and  e~mi*  is  an  integrat 
ing  factor  (§  13).  Hence 

e~mixu  =  Ce-mfXdx  -f  c,  or  u  =  ems  Ce'^Xdx  -f-  cem?  ; 
i.e.  (D  -  m^)(D  -  m3)y  =  emix  Ce~mixXdx  -f  ^Mix. 

Now  let  (D  —  ms)y  =  v. 

Then  (D  -  m^)v  =  ems  Ce~msX  dx  +  ^miz. 

This   is    also   linear  and  of  the  first   order,    hence   an    integrating 
factor  is  e~™"?.     Introducing  this,  we  have 


or 


vc-m?=  fX-™2H    Ce-m?X<ix  \ 
J  \J  J 

v  =  em?  Ce(^-m^\    (e-m?X 
J  \_J 


m  — 


JUT  LINEAR,  WITH   CONSTANT   COEFFICIENTS  99 

Hence    (D  -  m3)y  =  em*x  Ce(mrm^>x     Ce~mixX  dsc\d*  +  c"em<*  +  c'emf, 


where  c"  = 


m-,  — 


This  is  again  linear,  and  an  integrating  factor  is  e  m*x.     Using  this 
we  have 

yc-m&=  fX-'V*  {  Ce^r^A  Ce~mixXdx  \dx  \  dx 


=  em*x  Ce(m*-m*x  I  rV"!""^     Ce-mfXdx  \dx  \ 


dx 


This  law  in  the  case  of  the  nth  order  is  obvious  now.*     It  is 
I.  y  =  emnx  r^n-i-"1/*)*  p  •  .  Ce^i-™***  (  e~ 

•••  +cnemnx. 


Remark.  —  In  the  second  line  we  have  the  complementary  func 
tion,  with  which  we  are  already  familiar  (§  43).  (Let  the  student 
show  that  in  the  case  of  repeated  roots  of  the  auxiliary  equation  this 
method  leads  to  the  same  result  as  §  44.)  In  the  first  line  we  have 
the  particular  integral,  whether  the  roots  of  the  auxiliary  equation  are 
all  distinct  or  not. 

*  To  prove  this,  we  simply  need  assume  it  for  the  «th  order,  and  show  that  it  holds 
for  the  («  +  i)st  order.  This  can  be  done  at  once,  and  will  be  left  as  an  exercise 
to  the  student. 


ICO  DIFFERENTIAL   EQUATIONS  §47 

d*y      dzy        dy 

Ex.  i.    —  — —  2  -  —  =  e    . 

dx*      dx~         dx 


The  auxiliary  equation  is 

ftp  —  ;/z2  —  2  ;;/  =  o.     /.  m  —  o,  —1,2. 
The  complementary  function  is    Y=  cl  -f-  c#~x  +  ^"s 

The  particular  integral  is 


=^  Ce~3x  Cf(-  e-)(dx)*  =  -  e-x  Ce~34  £  dx\dx 


=  -xe-x  +  -e~x. 
39 

Since  e~x  is  already  part  of  the  complementary  function,  it  will  be 

sufficient  to  use  -  xe~x,  thus  giving  the  general  solution, 
«5 

y  =  fi  +  c*~*  +  W2x  +  -  xe~x. 

Ex.2.    (^2  +  3Z>  +  2)j^  =  ^. 

Ex.  3.    (Z>3  +  3  D*  +  3  D  +  i)j  =2e~x-  xze~x. 

Ex.  4.    (D1-  D-  2}y  =  sin  Jtr. 

E,5.  ^_=__. 


§43  LINEAR,    WITH    CONSTANT  COEFFICIENTS  IOI 

48.   Another  Method  of  finding  the  Particular  Integral.*  —  The 

general  method  of  finding  the  particular  integral  given  in  §  47  is 
frequently  long.  At  times,  the  first  integration  is  readily  obtained, 
but  the  successive  ones  are  long  and  tedious.  In  such  cases  the 
following  method  applies  : 

Starting  with  (D  —  m^) (D—m^-"(D— mn)y  =  X, 

we  can  write  symbolically 

y= 


(D- 
wheje  ' -  is  the  symbol  of  the  operation 


inverse  to  (D  —  m^(D—m^  •••  (D  —  m,^.  Just  as  sin"1  x  means  such 
a  function  of  x  that  when  we  operate  on  it  with  the  operator  sin  we 
get  x,  so  if  we  operate  on 


(D  - 

with  (D—m^(D—m^)  •••  (D  —  mn),  we  get  X.     Now  we  have  seen 
that  the  operator  (D—  m^)(D  —  ;«2)  •••  (D  —  mn)  is  equivalent  to  the 
successive    performance    of  the    operators    (D  —  m^,   (D  —  m2),  •••, 
(D  —  mn)  ;  and  besides,  the  order  of  the  latter  is  not  essential. 
Looked  upon  algebraically  the  fraction 


(D  -  m^(D  -  m2)  ...•(/>-  mn) 

is  equal  to  the  sum  of  the  partial  fractions 


a\        [        ^2       i    ...    , 


if  the  roots  of  the  auxiliary  equation  are  distinct. 

*  This  was  first  published  by  Lobatto,  Thtorie  des  Caracteristiqucs,  Amsterdam, 
&yj.  Independently  it  was  given  by  Boole,  Cambridge  Math.  Journal,  ist  series, 
Vol.  II,  p.  114. 


102  DIFFERENTIAL   EQUATIONS  §48 

Looked  upon  as  operators,  this  equality  still  holds  ;  for  to  verify 
the  equality  we  operate  on  both  with  (D  —  m^(D  —  m2)  •••  (D  —  /#„). 
Since  the  order  in  which  we  operate  with  these  factors  is  immaterial, 
the  result  will  be  that  all  the  operators  resulting  are  polynomials, 
which  can  be  treated  as  algebraic  expressions.  Hence  the  algebraic 
equality  of  the  symbolic  representatives  of  the  two  operators  means 
the  equality  of  the  operators,  and  the  original  operators,  in  fractional 
form,  are  also  equivalent;  i.e. 


x=    "*    x+    "2    x+  ... 

X. 


D  —  m 


If  we  put  u  —  — - — Xy  then  (D  —  m)u  =  aX. 


Integrating  this  linear  equation,  we  have  ue~mx  =  a  \  e~mxXdx,  or 

u  =  aemx  I  e^^Xdx.     Hence  the  particular  integral  may  be  put  in  the 
form 

/c  r 

e~mfXdx  +  a<>em*x  \  e~m*xXdx  +  •••  +  anemnX  I  e~mnxXdx. 
J  J 

Remark  /.  —  This  method  leads  to  a  real  particular  integral,  even  in  case  a  pair 
of  the  roots  of  the  auxiliary  equation  are  conjugate  complex  quantities,  a  +  i(i 
and  a  —  z'/3.  In  breaking  up  —  - —  into  a  sum  of  partial  fractions,  we  know 

that  the  sum —  H ~r^ r- r    ^s   equa^   to   — — — — ,    in 

which  k  and  /  are  real.     Hence  a\  and  a<i  are  also  conjugate  complex  quantities, 
say  X  -f  i/j.  and  X  —  tfj,. 


Now 


§49  LINEAR,  WITH  CONSTANT  COEFFICIENTS  1 03 

Since   ^  — X  may  be  gotten  from  the  above  by  changing  the  sign  of 

i  wherever  it  occurs,  the  two  have  the  same  real  parts,  while  their  imaginary 
parts  are  equal  but  of  opposite  signs.  Hence  their  sum  is  equal  to  twice  the 
real  part  of  either;  i.e. 


=  2  eax  (X  cos  PX-H  sin  0*)  (V«*  X  cos  0x  dx 
JJ 

+  2  eax  (X  sin  /3*+yu,  cos  px)  \  e~ax  X  sin  px  dx. 


Remark  2.  —  In  case  a  root  is  repeated  the  following  obvious  modification 
is  necessary  : 
To  fix  the  ideas  we  shall  suppose  one  of  the  roots,  m\,  is  a  triple  root.     The 

corresponding  partial  fractions  will  be  -  ?!  --  L  -  ^  --  1  --  ^  -  .  and 

D  -  wi  ^  (D  -  m-^*      (D  -  mi}z 

the  corresponding  terms  of  the  particular  integral  will  be  (I,  §  47) 

(  e~mf  X  dx  +  a2emix  (  (  e~mix  X  (dx}*  +  asemix  (  (  f  f~mtx  X 

Ex.1.     (Z)2- 

Ex.2.    (Z>3- 

Ex.  3.    (jy  +i)y  =  sec  x. 

Ex.4.     jy 


49.  Variation  of  Parameters.*  —  Another  general  method  of  ob 
taining  the  particular  integral,  known  as  the  method  of  variation  of 
parameters,  at  times  applies  very  readily,  especially  if  the  order  of  the 
equation  is  not  high.  The  method  consists  in  considering  the  con 
stants  in  the  complementary  function  no  longer  as  constants,  but  as 
undetermined  functions  of  x  such  that  when  substituted  in 
we  get  X,  and  not  zero,  as  is  the  case  when  they  are  constants. 

*  This  method  is  due  to  Joseph  Louis  Lagrange  (1736-1813). 


104  DIFFERENTIAL   EQUATIONS  §49 

Since  we  have  n  functions  at  our  disposal,  and  only  one  condition 
to  impose  upon  them,  it  is  clear  that,  theoretically  at  least,  we  can 
satisfy  this  requirement  in  an  indefinite  number  of  ways,  by  imposing 
any  other  n  —  i  conditions  we  please.  In  actual  practice  we  shall 
impose  these  conditions  in  such  a  way  as  to  simplify  our  work  as 
much  as  possible. 

The  method  will  be  carried  out  in  the  case  of  an  equation  of  the 
third  order.  (The  argument  will  readily  be  seen  to  apply  to  any 
order.) 

Let  the  equation  be 


(i)  ( 

and  let  the  complementary  function  be 


We  shall  try  to  find  tlt  c^  c8)  such  that  (2)   shall  be  a  solution  of 
(i).     This  still  allows  us  to  impose  two  conditions  upon  clt  c2)  ca. 
Differentiating  (2),  we  get 

—l 


dx  dx  dx 


Dy  = 

fix  dx  fix 


We  shall  now  use  one  of  the  two  conditions  at  our  disposal  by 
letting 

=o 


so  that  we  have 

(4)  Dy  = 


*  If  any  of  the  roots  of  the  auxiliary  equation  are  repeated  or  imaginary,  the  re 
sulting  change  in  the  form  of  the  complementary  function  causes  no  difference  in  thq 
process. 


§49  LINEAR,  WITH  CONSTANT  COEFFICIENTS  1 05 

Differentiating,  we  get 


T^vO  O  ~«    ~         .  9  »M     *«•         i  9  vn     i>        i  -wi    -».l^6l          ,  *«    vtt'Cf) 

Jjfy  =  ntfc-J*?  +  m*Ctf!*r  -\-  m^czem^x  +  m\i!*& — *  +  m<>emf  — = 

dx  dx 


Here  again  we  put 


(5)  m^       +  m^f       +  mj***       =  o, 

dx  dx  dx 


thus  using  the  second  condition  still  at  our  disposal,  so  that 
(6)  D2y  =  m,  VlX  +  mfc^f  +  mfcj***. 

Differentiating  again,  we  get 


(7)         JJ-y  =  mfctf™?  +  mfcrf"*  +  m/Caf***  +  ^i2^1       +  mff*? 

dx  dx 


. 
dx 

Substituting  (2),  (4),  (6),  (7),  in  (i),  and  remembering  that  with 
fit  £2,  £3  constant,  (2)  is  the  complementary  function,  we  have 

(8)  k#n{<r?  ^  +  k,m?e™?  ^  +  ktfnfS**  ~*  =  X. 

dx  dx  dx 

Equations  (3),  (5),  (8)  are  three  linear  equations  sufficient  to  de 

termine  —  >  —  2>  -^,  and  by  quadratures  clt  czt  c^  will  be  found  such 
dx    dx     dx 

that  (2)  will  be  a  solution  of  (i),  the  constants  of  integration  giving 
us  again  the  complementary  function. 

The  method  of  variation  of  parameters  applies  to  all  linear  equations,  whether 
the  coefficients  are  constants  or  not.  (Thus,  see  Ex.  4,  §  53.)  As  an  illustra 
tion,  we  shall  solve  the  general  linear  differential  equation  of  the  first  order 
(§  13)  by  this  method. 


106  DIFFERENTIAL  EQUATIONS  §49 

The  general  equation  is 

(1)  %-+Py=Q, 

dx 
where  P  and  Q  are  functions  of  x.     Let  us  consider,  first, 

(2)  dfx+Py  =  o,  or 

*•+/»*«.* 

.T 

Integrating,  we  get  log  jy  +  f  Pdx  =  C,  or 

(3)  ?*!"•=  fc  =  ft 

Now  let  <r  be  considered  a  function  of  x.      We  shall  determine  it,  so  that  (3) 
shall  satisfy  (i).     Differentiating,  we  have 


Comparing  this  with  (i),  we  see  that  we  must  have 

dc  ~    f  Pdx  r         [Pdx    ,  . 

—  =  QeJ       ,  or  c=  I  Qe}       dx  +  c. 
dx  J 

:.yfdx=   \  Qe*Fd*  dx  +  f1  is  the  solution  (the  result  we  obtained  in§  13). 
Ex.  1.     -0  +y  =  sec  x.     (Ex.  3,  §  48.) 

'The  roots  of  the  auxiliary  equation  are  ±  i.     Hence  the  comple 
mentary  function  is 


y  =  d  cos  x-\-  <:2  sin  x. 
Dy  =  —  c±  sin  x  +  c%  cos  #  -4- 


-2 


—  ^  cos  JP  —  <r2  sin  x  —  sin  ,*  — ^ ^  +  cos  je  —  • 

dx  dx 


§50  LINEAR,  WITH   CONSTANT  COEFFICIENTS  IO/ 

Substituting  in  the  differential  equation,  we  have 


+  GQIM-  sec  x. 

dx  dx 


Besides,  we  chose  cos  x  —  -  -f  sin  x  —  -  =  o. 
dx  dx 


dx  cos  x 

~dx  =  l' 
And  the  complete  solution  is 


y  =  d  cos  x  +  C2  sin  ^  4-  cos  #  log  cos  x  -f-  #  sin  #. 

_,  d^y 

•^x*  *•  ~7~9  >y  ~  *an  ^ 

dk2 

50.  Method  of  Undetermined  Coefficients.  —  We  shall  conclude  the 
discussion  of  the  problem  of  finding  the  particular  integral  with  an 
account  of  a  method,  which,  while  not  applicable  in  all  cases,  is 
relatively  simple  whenever  it  can  be  used.  It  applies  to  all  cases  in 
which  the  right-hand  member  contains  only  terms  which  have  a 
finite  number  of  distinct  derivatives.  Such  terms  are  x*t  ejx,  sin  mx9- 
cos  nx,  and  products  of  these,  where  h  is  any  positive  integer,  and 
/,  m,  n  are  any  constants. 

By  this  method  we  find  the  particular  integral  U  by  inspection,  or 
by  trial,  as  it  were. 

If  we  take,  as  a  first  trial,  the  terms  of  the  right-hand  member  X, 
each  prefixed  by  an  undetermined  multiplier,  we  shall  find  that,  as 
a  rule,  on  substituting  this  in  the  left-hand  member,  f(D)y>  other 


108  DIFFERENTIAL   EQUATIONS  §50 

terms  arise  as  a  result  of  differentiation.  Consequently,  we  shall 
use  for  U  the  sum  of  all  the  terms  of  X,  together  with  all  those 
arising  from  them  by  differentiation  (by  hypothesis  there  are  only 
a  finite  number  of  these),  each  prefixed  by  an  undetermined  mul 
tiplier.  We  then  equate  identically  to  X  the  result  of  this  substitu 
tion  (i.e.  we  equate  the  coefficients  of  corresponding  terms).  This 
will  give  as  many  equations  of  condition  among  the  undetermined 
multipliers  as  there  are  distinct  terms  'mf(D)  U.  This  number  is 
either  equal  to  or  less  than  the  number  of  undetermined  multipliers 
(for  all  the  terms  obtainable  from  U  by  differentiation  need  not 
occur),  and  these  multipliers  can  then  be  calculated. 

Ex.1.      --^  +  4y  =  x2+cosx. 
dx* 

The  roots  of  the  auxiliary  equation  are  ±  2  /. 
.*.  Y=  A  cos  2  x  +  B  sin  2  x. 

For  a  particular  integral  we  take, 

U  '—  ax2  -+-  bx  +  c  -\-/cosx  +  gsinx. 

Then,  IPU=  2  a  —  /cosx  —  gsinx. 

.\f(D)  (7=  4  ax*1  4-4^  +  2^  +  4^+  3/cosjc  -f 


Equating  coefficients  of  this  to  those  of  X  we  have, 
4  #  =  i  . 

4  b  =  o 
2  a-\-  4  c  =  o 


3/=i  f=~, 

«J 

3^=0  £-=0. 


§50  LINEAR,  WITH  CONSTANT  COEFFICIENTS  IOQ 

Hence  the  general  solution  is 

y  =  A  cos  2  x  +  B$\n.2X-}--x2  —  -  +  —  cos  x. 

4          83 


Ex.2. 

The  complementary  function  is  readily  seen  to  be 


To  find  the  particular  integral  it  will  be  simpler  to  replace  —  sin2.* 
by  -  cos  2  x  —  -.     Doing  this,  we  put 

2  2 


-\-  be**  -f-     <rcos2#-f-    /sin  2 

.•.  DU—  2  axe"2*  +  (a  +  2  <£)<?2x  -+-  2/cos  2  Jt:  —  2  c  sin  2  .*:. 
D~U=  4  a^x  +  4  («  +  ti)  e*x  —  4  c  cos  2  jc  —  4/sin  2  x. 
f(D)U=  axe1*  +  (2  a  +  %2x  -  (3  c  +  4/)  cos  2  # 

-  (37—  4  ^)  sin  2  ^  +^i 
Hence  we  must  have 

a  =  2  .-.  a  =  2, 

2  a-\-  b  =  o  ^  =  —  4, 


50 


and  the  general  solution  is 


y=(cl-\-  czx)  e*  +  2  x<?*—  4  e1*  —  -3-  cos  2  #  --  —  sin  2  jr  —  -  • 

50  25  2 


IIO  DIFFERENTIAL   EQUATIONS  §50 

Ex.3. 


Ex.  4.    (Z>2  +  2  D  +  i)  jy  =  3^  -  cos  #. 
Ex.5. 


This  method  will  be  in  default  in  either  of  the  following  two  cases  : 
i°  If  a  term  in  the  right-hand  member  is  also  a  term  in  the  com 

plementary  function,  it  is  clear  that  the  substitution  of  such  a  term 

or  any  of  its  derivatives  for  y  \nf(D*)y  will  not  give  rise  to  that  term. 

As  a  matter  of  fact  we  get  zero. 

We  shall  first  suppose  that  the  root  of  the  auxiliary  equation,  to 

which    the    term   in   question   corresponds,  is   a  simple   one;    if  u 

is  the  term,  this  amounts  to  having 

f(D)u  =  o,  but/'(Z>>  =£  o, 


Now  since  f(D)  is  a  polynomial  in  D,  and  since  Dk(xu}  = 
"^u  it  is  clear  fo&f(D)(xu)  =  xf(D)u+f(D)u. 

Since  f(D]u,  by  hypothesis,  is  different  from  zero,  it  follows  that 
if  for  y  in  f(D)y  we  substitute  xu  +  terms  derived  from  this  by 
differentiation,  we  shall  have  resulting  the  term  u  -f  terms  arising 
from  it  by  differentiation,  and  none  other. 

Perfectly  generally,  if  the  term  u  is  also  a  term  in  the  complemen 
tary  function  which  corresponds  to  an  r-fold  root  of  the  auxiliary 
equation,  then 

f(D)u  =  o,/'(/?)»  =  o,  ...,/"-1>(Z>)»  =  o,  but/(r)(Z>>  gfc  o. 

*  Letting  m  be  the  root  corresponding  to  u,  we  have  (§  31)  that/'(w)  =£0  if  m  is  a 
simple  root  of/O)  =  o.  Hence  u  is  not  an  integral  of  the  equation  f(D)y  =  o. 


§50  LINEAR,  WITH   CONSTANT  COEFFICIENTS  lit 

Since 


and  since  /  (Z>)  is  a  polynomial  in  Z>  with  constant  coefficients,  it 
follows  that 


All  of  the  terms  on  the  right  are  zero,  except  f(T\U)u,  which  is 
definitely  not  zero.  Hence  if  for  y  in  f(D)y  we  substitute  .#r#  + 
terms  arising  from  this  by  differentiation,  we  shall  obtain  the  term 
u  +  terms  arising  from  it  by  differentiation,  and  none  other. 

2°  The  second  case  where  the  original  method  will  be  at  fault  is 
where  terms  of  the  type  tfu  occur,  u  being  a  term  in  the  comple 
mentary  function.  A  similar  modification  of  the  method  applies 
here.  Suppose  that  u  corresponds  to  an  r-fold  root  of  the  auxiliary 
equation.  *  As  before,  we  have 

-*f  (U)u 


2  ! 

(t+r)(t+r- 


*  Of  course  xu,  x2u,  •••,  xr~^u  are  also  terms  in  the  complementary  function.  In 
this  discussion,  u  is  supposed  to  be  that  term  which  does  not  contain  x  as  a  factor 
otherwise  the  exponent  /  would  be  indeterminate. 


112  DIFFERENTIAL  EQUATIONS  §50 

All  of  these  terms  on  the  right  are  zero,  except  f(r\D)u,  which  is 
definitely  not  zero.  Hence,  if  for  y  in  f(D]y  we  substitute  xt+ru  + 
terms  arising  from  it  by  differentiation,  we  shall  obtain  x1 u  +  terms 
arising  from  it  by  differentiation,  and  none  other. 

We  are  now  in  a  position  to  formulate  our  rule  : 

When  the  right-hand  member  of  the  differential  equation  contains 
only  terms  which  have  a  finite  number  of  distinct  derivatives,  take  for 
particular  integral  the  sum  of  all  the  terms  together  with  all  those  ob 
tained  from  these  by  differentiation,  prefixing  to  each  of  them  an  un 
determined  coefficient.  These  coefficients  are  determined  by  substituting 
the  trial  particular  integral  in  the  differential  equation  and  equating 
coefficients.  In  case  a  term  in  the  right-hand  member  is  a  term  in  the 
complementary  function  or  a  term  in  it  multiplied  by  an  integral  power 
of  x,  which  term  corresponds  to  an  r-fold  root  of  the  auxiliary  equa 
tion,  replace  that  term  in  the  right-hand  member  by  xr  times  it  in 
making  up  the  trial  particular  integral. 

Remark.  —  It  may  not  always  be  necessary  to  insert  all  the  terms 
suggested  by  the  general  rule.  These  can  frequently  be  detected  by 

inspection.  Thus  in  Ex.  i,  since  the  coefficient  of  •"-  in  the  differ 
ential  equation  is  zero,  the  terms  x  and  sin  x  in  the  trial  particular 
integral  are  unnecessary,  for  these  will  obviously  not  appear  as  a  re 
sult  of  substituting  ax2  +/  cos  x  in  the  equation.  If  any  unnecessary 
terms  are  put  in  the  trial  particular  integral,  that  fact  will  show  itself 
by  having  their  coefficients  turn  out  to  be  zero.  So  that,  excepting 
the  unnecessary  labor,  the  introduction  of  extraneous  terms  in  the 
trial  integral  is  not  serious.  It  is  also  obviously  useless  to  put  in  any 
terms  which  appear  in  the  complementary  function.  (If  such  a 
term  is  included  in  the  trial  particular  integral  its  coefficient  will,  of 
course,  not  appear  in  the  resulting  equations  among  the  coefficients. 
This  means  that  this  coefficient  may  be  chosen  arbitrarily ;  which  is 
exactly  as  it  should  be.)  As  a  consequence,  when  any  term  in  the 


§51  LINEAR,  WITH  CONSTANT  COEFFICIENTS  113 

right-hand  member  is  replaced  by  xr  times  it  in  either  of  the  two 
exceptional  cases  referred  to  in  the  rule,  only  those  terms  obtained 
from  this  by  differentiation  which  are  not  in  the  complementary 
function  need  be  added. 

Ex.  6.    (IP  -  2  D*  -  3  D)y  =  3  x2  -f  sin  x. 

Hint.  —  Since  o  is  a  simple  root  of  the  auxiliary  equation,  and  i  is 
therefore  a  term  in  the  complementary  function,  we  shall  have  to  try 
ax?  +  fix2  -f  ex  to  get  x2.  Moreover,  sin  x  is  not  a  part  of  the  com 
plementary  function.  Hence  the  trial  integral  is  U—  ax3  -f-  bx*  4- 
cx  -f-/  sin  x  +  g  cos  x. 


Ex.7. 

Ex.8.    (D2-2  D}y  = 

Ex.9. 


51.    Cauchy's  Linear  Equation.  —  The  linear  differential  equation 

=  X* 
x  x 

where  the  coefficient  of  —  2-  is  a  constant  times  xr,  is  at  once  reduc- 
dxr 

ible  by  the  transformation  x  =  e*  to  an  equation  with  constant  co 
efficients.     For 

dy  _  dy  dz  _  i  dy 
dx      dz  dx      x  dz  ' 

d*y  _  i  (d*y      dy\ 

~~7~9  —  —     5i  "TV       ~T   ' 

ax*  x-  \av       dz) 

*  This  form  of  linear  equation  is  often  called  the  homogeneous  linear  equation.  This 
seems  rather  unfortunate.  I  prefer  to  reserve  this  name  for  any  linear  equation  which 
is  homogeneous  my  and  its  derivatives,  in  conformity  with  a  large  number  of  writers 
on  the  subject,  and  shall  refer  to  the  above  linear  equation  (i)  as  the  Cauc/iy  linear 
differential  equation,  after  Augustin  Louis  Cauchy  (1789-1857^.  See  his  Exercises 
d'  Analyse. 


114  DIFFERENTIAL   EQUATIONS  §51 


dz 


dny  _ 
dxn~ 


or  if  we  let  —  =  Jfry,  we  have 
dz 

xDy    =  &y, 


and  (i)  becomes 

(2) 


where  Z  is  what  ^f  becomes  as  a  result  of  the  transformation.* 
(2)  is  obviously  a  linear  equation  with  constant  coefficients. 
More  generally,  the  equation 

£o(*  +  ^)n|^  +  £i(tf  +  ^^ 

is  readily  seen  to  be  reducible  to  a  linear  equation  with  constant  co 
efficients  by  the  substitution  a  +  bx  =  e*. 

*  For  another  general  method  of  solving  a  Cauchy  linear  equation  see  footnote,  $  74. 
t  This  form  of  the  linear  equation  is  referred  to  as  Legendre's  linear  equation,  after 
Adrien  Marie  Legendre  (1752-1833). 


§5*  LINEAR,  WITH   CONSTANT  COEFFICIENTS 

Ex  1     x^    ^  -f-  x       v  =  x  log  x 

dx&        dx 

Putting  x  =  e*,  this  becomes 

The  roots  of  the  auxiliary  equation  are  i,  i,  i. 

Hence  the  complementary  function  is  Y=  (^  +  c& 

In  this  case  method  I  (§  47)  gives  the  particular  integral  at  once. 

(*  f*  f*  z^ 

We  have  U=ez  \    \    I  e~zzez(dzf=  ez  — . 

*J  *s   */  24 


.'.The  solution  is y  =  (c±  +  czz  -f 
or  y  =  [/!  +  ^2  log  x  +  ^3  (log  ^)2] 

Ex.  2.    (^jD3  +  2  ^2Z>2  +  2) y  =  10 f#  +  -\ 

V         ^ 


Ex.3. 


_       __.. 

(I  —  X) 

Ex.4.     ^I2_^I  6 


52.  Summary. — The  problem  of  solving  a  linear  differential  equa 
tion  consists  of  two  parts,  the  finding  of  the  complementary  function, 
and  the  finding  of  a  particular  integral  (§42). 

The  finding  of  the  complementary  function  in  the  case  of  an  equa 
tion  with  constant  coefficients  /(Z>)_y—  X  is  simply  an  algebraic 
problem,  viz.  the  solution  of  the  equation/ (m)  =  o.  According  as 


Il6  DIFFERENTIAL  EQUATIONS  §52 

the  roots  are  distinct  and  real,  repeated  or  complex,  the  complemen 
tary  function  takes  one  of  the  forms  indicated  in  §§  43,  44,  45.  The 
problem  of  finding  the  particular  integral  may  be  attacked  by  any  of 
the  four  methods  given  in  §§  47,  48,  49,  50. 

An  estimate  of  the  relative  merits  of  these  methods  may  be  summa 
rized  as  follows :  The  methods  of  §§  47  and  48  (which  will  be  referred 
to  as  I  and  II  respectively)  and  that  of  variation  of  parameters  (§  49) 
have  the  advantage  of  absolute  generality.  But  as  is  usually  true  in 
such  cases,  the  actual  carrying  out  of  these  methods  is  frequently 
very  long  and  laborious.  Excepting  in  certain  cases,  soon  learned 
by  experience,  method  II  is  simpler  than  I,  in  that  it  requires  several 
integrations  of  the  same  kind  instead  of  several  successive  integra 
tions.  The  method  of  variation  of  parameters  has  the  great  advantage 
of  being  readily  retained  in  mind,  but  is  frequently  long  and  labori 
ous,  especially  if  the  equation  is  of  higher  order  than  the  second. 
The  method  of  undetermined  coefficients  (§  50),  although  not  abso 
lutely  general,  applies  to  a  very  large  number  of  cases  that  actually 
occur.  In  such  cases  where  it  does  apply  it  has  the  advantage  of 
involving  only  the  operations  of  differentiation  and  the  solution  of 
simultaneous  linear  algebraic  equations.  Integration  is  not  involved. 
Besides,  it  is  very  readily  retained  in  mind.  The  actual  work  of 
carrying  out  this  method  is  straightforward  and  not  difficult.  It 
may  at  times  be  long,  but  usually  it  is  no  longer  than  the  other 
methods,  if  as  long. 

As  a  rule,  then,  whenever  the  method  of  undetermined  coefficients 
applies,  it  is  probably  the  most  desirable  one  to  use.  An  instance 
of  an  exception  to  this  is  illustrated  by  Ex.  i,  §51.  [Generally  we 
may  say  that  the  method  I  is  preferable  in  case  the  right-hand 
member  contains  a  term  elx  or  ^xf(x),  where  f(x)  can  be  integrated 
readily  any  number  of  times,  and  when  the  auxiliary  equation  is 
(m  —  /)r  =  o.]  If  it  is  obvious  on  inspection  that  different  methods 
apply  most  readily  to  the  various  terms  in  the  right-hand  member, 


§  52  LINEAR,    WITH   CONSTANT  COEFFICIENTS  1  1  7 

employ  the   method   that  is  simplest  for  each  term  and   take   the 
sum  of  the  results.     This  is  true,  for  instance,  in  Ex.  1  1  below. 
The  equation 


(including  as  a  special  case  the  Cauchy  equation  where  a  =  o,  b  =  i) 
is  reducible  to  one  with  constant  coefficients  by  the  substitution 
a  +  bx  =  c*  (§51). 

Ex.    1.    (£>2-$  £>  +  6)y  =  cosx-e2x. 

Ex.    2.    (Z>4  —  i  )y  =  ex  cos  x. 

Ex.    3.    (Z>2+2£>  +  i)y  =  2  Xs  — 

Ex.    4.    (D  +  i)3j  =  xe~*. 

Ex.    5. 

Ex.    6. 

Ex.    7. 

Ex.    8. 

Ex.    9. 

Ex.10.    (Z>2  +  i)y  =  sec2  jc. 

Ex.11.    (Z)-i)3>'  =  ^ 

Ex.12.    (&-&-  3 

Ex.  13.     Z>2  -h  i'  =  x  cos  #. 


Ex.14.    (jc3^>3+2^2Z>2-^Z)+i)7--- 

2 


Ex.  15.    (Z)3  —  i  )y  =  xex  +  cos2  x. 

Ex.  16.    (Z>  -  i)2y  =  cos  jc  +  e*  +  s*e*. 

Ex.  17.    Study  the  motion  of  a  simple  pendulum  of  length  /  and 
mass  m  swinging  in  a  vacuum. 


Il8  DIFFERENTIAL  EQUATIONS  §52 

The  only  force  acting  is  gravity ;  it  acts  vertically  downward,  and 
its  intensity  is  —  mg.  If  s  represents  the  length  of  arc  measured  from 
the  lowest  point  of  the  pendulum,  then  at  any  moment  when  the 
pendulum  makes  an  angle  6  with  the  vertical,  s  =  IB,  and  the  accelera 
tion  is  /  — -•  The  component  of  the  force  of  gravity  along  the  tan 
gent  to  the  path  is  —  mg  sin  0.  Hence  the  equation  of  motion  is 

ml  — 2  =  —  mg  sin  0. 

If  0  remains  very  small  throughout  the  motion,  we  may  replace  sin  0 
by  0  as  a  first  approximation.  Our  equation  then  takes  the  form 


[This  is  the  differential  equation  of  simple  harmonic  motion.] 
Solving  this,  we  have  0  =  A  cos  f  */£/  +  B  \ 

where  A  and  B  are  constants  depending  on  the  initial  value  of  0  and 
of*. 

dt 

A  determines  the  amplitude,  while  B  determines  the  phase. 

The  period  is  *»\/p  *.'•  the  state  of  motion  will  be  identically  the 

same  for  two  values  of  /  whose  difference  is  an  integral  multiple  of  this 
quantity. 

Ex.  18.   Consider   the  case  of  a   simple   pendulum  moving  in  a 
resisting   medium  where   the  resisting  force   is  proportional  to  the 

velocity,  say  —  2  km  —  • 


§52  LINEAR,  WITH  CONSTANT  COEFFICIENTS  IIQ 

Putting^  =  «2,  the  differential  equation  to  be  solved  is 


[The  same  equation  arises  in  the  case  of  damped  vibrations  of  the 
needle  of  a  galvanometer.] 

Ex.  19.  In  the  case  of  forced  vibrations,  such  as  when  a  magnet 
is  brought  up  periodically  to  a  vibrating  tuning  fork,  the  equation  of 
motion,  in  case  there  is  no  resisting  force,  is 


(a)  +  n 

(the  cases  when  m^n  and  m  =  n  must  be  distinguished). 

If  the  resisting  force  is  proportional  to  the  velocity,  the  equation 
of  motion  is 

0)  ^.+  8**+4M*.CoMff* 

ar          at 

Ex.  20.  A  particle  is  projected  with  velocity  VQ  away  from  the 
center  of  an  attractive  force.  If  the  acceleration  of  the  particle  due 
to  the  force  is  proportional  to  the  distance,  find  the  motion. 

Ex.  21.  If  in  Ex.  20,  the  force  is  a  repellent  one,  and  the  particle 
is  projected  toward  the  center  of  force  with  the  velocity  VQ,  find  the 
motion. 

Ex.  22.  Find  the  motion  of  a  heavy  particle  moving  without  fric 
tion  along  a  massless  straight  line  which  rotates  about  one  of  its 
points  in  a  vertical  plane  with  constant  angular  velocity.  The  only 
force  acting  is  gravity. 


I2O  DIFFERENTIAL    EQUATIONS  §52 

If  r  is  the  distance  of  the  moving  point  from  the  point  about 
which  the  line  rotates,  and  if  o>  is  the  angular  velocity  of  the  line, 
Lagrange's  equation  (generalized  coordinates)  is 


o 
—  —  coV  =  —  g  sm  o>/. 


Ex.  23.  If  a  condenser  of  capacity  S,  charged  with  a  quantity  of 
electricity  Q,  is  introduced  into  an  electric  circuit,  it  will  discharge 
by  sending  a  current  through  the  circuit.  If  q  is  the  quantity  of 
electricity  in  the  condenser  at  any  instant  during  the  discharge,  it 
will  be  determined  by 

~dt^~L  ~dt  +  ~LS  =  °' 
where  L  is  the  self-inductance  and  R  the  resistance  of  the  circuit. 

Here  the  auxiliary  equation  is 


=  o. 


R 
— 


To  determine  A  and  B  we  make  use  of  the  fact  that  when  /  =  o, 
q  =  Qy  and  - — -  =  i  =  o,  where  /  is  the  current ;  i.e. 

A  +  B  =  Q,  and  miA  +  mzB  =  o, 


§52  LINEAR,  WITH   CONSTANT   COEFFICIENTS  121 


i  A  wi>>.         rt         m\\  j 

whence  A  —  --  ^—  ,  B  —  -  -^^-,  and 

m±  —  mz  m\  —  m2 

q  —  -  ^  —  (m^J  —  m^S)  , 
ml  —  m2 


dt     m±  —  mz 

Noting  the  values  of  m±  and  m2,  we  see  that  q  and  i  diminish  con 
tinually,  but  do  not  become  zero  for  a  finite  value  of  f,  although  they 

r> 

are   practically   negligible   very  soon  when  —  is  a   large   quantity, 
which  it  usually  is. 

2°    If  &  =  *, 


To  determine  A  and  B  we  have, 

A  =  Q,    *-A-3  =  o,   or^     -, 

2  L  2  L 

O  -^-t 

whence  q  =  -^  (2  L  4-  Rf)e  u  , 

2  L 


Here  again  ^  and  /  diminish  rapidly,  without  vanishing  for  a  finite 
value  of  /,  although  they  are  soon  negligible  as  a  rule. 


=  a  ±  /j8. 

,\  ^  =  <fa'(^  cos  fit  +  B  sin  /?/),  and 


cos 


122  DIFFERENTIAL  EQUATIONS 

To  determine  A  and  B  we  have, 

A  =  Q,  a. 
whence 


§5* 


P 


=  Qe 


LS     4Z2 


Both  ^  and  /  are  periodic  functions  of  period  T= 


2  7T 


=55  »    S0 


that  sometimes  they  are  positive  and  sometimes  negative.  The 
amplitude  in  either  case  is  a  constant  times  e  ~2Lt,  which  usually 
diminishes  very  rapidly  with  /.  But  in  specially  constructed  circuits 
in  which  R  is  small  relative  to  Z,  an  oscillatory  discharge  may 
be  realized.  —  I.  C.  and  J.  P.  JACKSON,  Alternating  Currents  and 
Alternating  Current  Machinery. 


CHAPTER   VIII 
LINEAR  DIFFERENTIAL  EQUATIONS  OF  THE  SECOND  ORDER* 

53.  Change  of  Dependent  Variable.  —  While  the  problem  of  solv 
ing  linear  differential  equations  of  the  first  order  can  always  be  carried 
out  (that  is  to  say,  we  can  reduce  it  to  one  of  quadratures,  §  13), 
that  of  solving  equations  of  the  second  order  can  be  carried  out  in 
only  a  comparatively  small  number  of  cases. 

The  general  type  of  a  linear  equation  of  the  second  order  is 


where  P,  Q,  X  are  functions  of  x  only. 

Let  us  try  the  following  change  of  dependent  variable, 
(2)  y  =  y&. 

r™  dv         dv  ,  dy,  d2y         cPv  ,      dy,  dv  , 

Then     •/  =  yl  —  +  -^  v,      --+  =yl  —  +  2  -fi  •—  +  - 
dx         dx      dx  dx2         dx1         dx  dx      rfx2 

and  equation  (i)  becomes 

/  \  d2v  ,    r,  dv  ,    ^          „ 


*  In  this  chapter  we  shall  consider  methods  which  apply  more  especially  to  linear 
differential  equations  of  the  second  order.  Of  course,  the  general  methods  of  the  next 
chapter  apply  to  linear  equations  of  the  second  order  as  well.  But  owing  to  the  gen 
eral  plan  of  solution  of  equations  of  higher  order  than  the  first  (§  56),  it  is  desirable  to 
have  available  the  methods  given  in  this  chapter. 

123 


124  DIFFERENTIAL   EQUATIONS  §53 


~ 

,  n        2  ay,   .    n       /-»       fi 

where        P^-^i  +  P,      Q1  = 


~7~       \LS  ^ 

fix  v      X 

,  X1  =  -- 

doc                                yl  yl 

Two  uses  may  be  made  of  this. 

i°  By  inspection*  or  other  means  a  particular  integral  of  the 
equation  when  we  put  X  =  o  may  be  known.  If  we  let  this  be  yl9 
we  have  Qt  =  o,  so  that  (3)  becomes 


~E3"r-l"7"  =      l* 

ay?         ax 

If  now  we  let  —=pt  we  have  a  linear  equation  of  the  first  order 
dx 


dx 
which  can  be  solved  for/  (§13).     A  quadrature  will  then  give  y. 


dx 


Here   x  is   a    particular    integral    (since   P=—  Qx).       Putting 
y  =xv,  we  have 


dx 


*  Thus,  for  example,  if  P  =  —  Qx,  x  is  evidently  such  a  particular  integral.  Again, 
if  I  +  P  +  Q  =  o,  ex  is  such,  or,  if  i  —  P-\-  Q=  o,  e~x  is  one  ;  or  more  generally,  it  may 
be  possible  to  note,  by  inspection,  a  number  m,  such  that  m2 -}-  Pm-{-  Q  —  o;  in  this 
case  emx  is  such  an  integral. 


§  53  LINEAR,  OF  THE  SECOND   ORDER  125 

r(--*2V*  ~ 

is         e    x          or  #v  3  • 


An  integrating  factor 


f!      r 

s  =  i 


or  P  =  ~r 

f      dx 


and  v  =  l  +  flCxr 

x        J 

whence  y  =  i  +  c\x  \  a 

.79..  J.. 

Ex.  2. 

Ex.  3.    I 
Ex.4,    (i- 


Here  x  and  ^  are  particular  integrals,  when  the  right-hand  mem 
ber  is  replaced  by  zero.  Hence,  by  property  A,  §  42,  the  comple 
mentary  function  is  c^x  +  c^x.  To  find  the  particular  integral  which 
must  be  added  to  the  complementary  function,  the  method  of  varia 
tion  of  parameters  (§  49)  may  be  employed.* 

*  Besides  the  method  of  variation  of  parameters  one  can  sometimes  use  with  facility 
a  general  form  for  the  particular  integral  given  by  Lie  in  his  Differ entialgleichungen, 
p.  429.  This  form  also  appears  in  the  author's  Lie  Theory  of  One-Parameter  Groups,  p.  174. 


126  DIFFERENTIAL   EQUATIONS  §53 

2°    If  we  put  />!  =  <>,/.*.       -^  +  />=0, 

yidx 
we  have  log  y1  =  —  -  J  Pdx,  or 

(4)  .n  =  HP* 

Using  this  value  for  ^  we  have 

Q1  =  Q-^-lp\    X^XeW**. 
2  dx      4 

Now  it  may  turn  out  that  O  -----  P2  is  a  constant,  in  which 

2  dx      4 

case  (3)  is  an  equation  with  constant  coefficients  ;  or  it  may  be  a  con 
stant  divided  by  x2,  in  which  case  we  have  a  Cauchy  equation,  and 
the  further  substitution  x  ==  <?*  will  reduce  it  to  one  with  constant 
coefficients  (§  51). 

Ex.  5.   sin  x  —2-  -f  2  cos  x  -2-  4-  3  sin  x  •  y  =  e*, 
dx*  dx 

or  -r?+2  cot  «*^r  +  3  j^^csc  x. 

dx*  dx 

Here  0-  *  —  -~/>2=  3  4-csc2*-  cot2*  =  4. 

2dx      4 

Hence  y  =  ve~^Pdx  =  v  esc  x  transforms  the  equation  to 


Integrating,  v  =  t\  cos  2  ^  +  ^2  sin  2  #  -f  -^", 

and  JK  =  ^i  (cos  ^?  cot  x  —  sin  .#)  +  c2  cos  jc  +  -  e*  esc  ^. 

*  This  result  can  be  written  at  once  without  actually  carrying  out  the  transformation. 


§54  LINEAR,  OF  THE  SECOND  ORDER  127 


Ex.6.          - 

do?  dx 


Ex.7. 


Ex.8.   s+r 

do?        dx 


54.   Change  of  Independent  Variable.  —  If  we  introduce  a  new  inds 
pendent  variable  z,  we  have 


_  =          _  =  , 

dx~  dz  dx    do?~  dz*\dx)       dzdof 


and  the  equation  (i)  becomes 


0-L.pfBL 

r-y    dx*    _Jx_dy  +  JL-         x 


It  may  happen  that  if  we  put      \   =  ±  i,  ?>.  —  =  V±  (?  (where 

fdz\  dx 

\dx) 

we  choose  that  sign  which  will  make  the  square  root  real),  the  co 
efficient  of  -2.  reduces  to  a  constant.     If  such  is  the  case,  our  equa- 
dz 

tion  (5)  is  linear  with  constant  coefficients,  and  can  be  solved  by  the 
methods  of  Chapter  VII. 


128  DIFFERENTIAL   EQUATIONS  §54 

Remark.  —  If  the  result  of  putting  •  =  ±  i  is  to  transform  the  equation 

\dx) 

into  —  2-  +  K  -%-  ±  y  —  o,   the   transformation  —  ^  —  =  ±  a,  where  a  is  any  con- 
d&  dz  d 


stant,  will  give  us  —£-  -f  Va  K  ~  ±  ay  —  o.      In  either  case  we  have  a  linear 
dfe2  </z 

equation  with  constant  coefficients.  But  if  K  involves  a  square  root  factor,  a 
may  be  so  chosen  that  V«  K  is  rational,  and  the  actual  work  is  thus  simplified. 
For  example,  see  Ex.  5  below. 


If 


Hence,  introducing  z  =  e*  as  the  new  independent  variable,  the  equa 
tion  becomes  j  * 


Its  solution  is          y  =  (^  +  ^r^)  ^~z  +  22  —  4  2  +  6. 
Replacing  0  by  its  value  in  terms  of  x,  we  have  finally 


Ex.2,    (i-^-         +  ^^o. 
<fo2         ^r 

Ex.  3.    ^  4-  tan  jtr^  +  cos2*  •  j  =  o. 


Ex.5. 

dx 

This  result  can  be  written  at  once  without  carrying  out  the  transformation. 


§55  LINEAR,  OF  THE   SECOND  ORDER  1  29 

55.    Summary.  —  There  is  no  general  method  for  solving  the  linear 

differential  equation  of  the  second  order,  —  —  +  P  —  +  Qy  =  X.    In 

dx?          dx 
actual  practice  we  proceed  as  follows  : 

i°.    If  by  inspection,  or  otherwise,  we  know  a  particular  integral 
yl  when   the    right-hand   member   is   made   zero,  then  y=y1v  will 

reduce  the  equation  to  a  linear  one  of  the  first  order  when  _  is  con 
sidered  a  new  variable  (§53,  i°). 

2°    If  such  a  particular  integral  is  not  known,  the  next  thing  to  do 

is  to  find  the  value  of  Q  —  -—  —  -P'2.     If  this  is  a  constant/or  a 

2  dx      4 

constant  divided  by  x2)  the  equation  is  reducible  to  one  with  con 
stant  coefficients,^*  to  a  Cauchy  equation)  by  substituting  y  =ylvt 
-and  then  it  fc  timf  fr^nlmlnt"  JiJ^jj^!  (§  53,  2°).* 

3°    If  the  previous  method  does  not  apply,  put  —  =  V±  Q  (using 

(tOC 

that  sign  which  will  make  the  square  root  real)  ;  then  substitute  in 

&.j.-j>* 

dx2  dx 

If  this  turns  out  to  be  a  constant,  the  method  applies, 


/  ,  \2  —  • 

(  — 
\dx) 


and  then  it  is  time  to  find  z  from  —  =  V  +  O  (§  ^4.}  * 

dx  ^  v    OHV* 


Ex.  1.   x 


Ex.2.    (*-3)-  (4*  -9) 

dx*  dx 


Ex.3. 


*  Emphasis  should  be  laid  on  the  fact  that  in  the  application  of  the  test  as  to 
whether  this  method  applies  no  integration  is  required.  It  is  only  after  one  is  assured 
the  method  worlcs  that  a  new  variable  need  be  sought. 


130  DIFFERENTIAL   EQUATIONS 

Ex.    4.    (*2  +  i) -^— 2  *--^  +  27  =  o. 
dx*  dx 

Ex.     5.    ^-(^-O^-H*-!),-* 


Ex.    6.   **-4X+(6  +  **) 7=0. 
tbr  dx 


Ex.    7. 


Ex.    8. 


Ex.    9. 


Ex.10. 


CHAPTER   IX 

MISCELLANEOUS  METHODS  FOR  SOLVING  EQUATIONS  OP 
HIGHER  ORDER  THAN  THE  FIRST 

56.  General  Plan  of  Solution.  —  There  is  no  general  direct  method 
for  solving  a  differential  equation  of  higher  order  than  the  first,  ex 
cepting  in  the  case  of  linear  equations  with  constant  coefficients  and 
those  reducible  to   such  (Chapter  VII).     The  general  plan  in  all 
other  cases  is  to  try  to  transfer  the  problem  to  that  of  solving  an 
equation  of  lower  order.     We  shall  consider  some  classes  of  equations 
for  which  this  can  be  done. 

57.  Dependent  Variable  Absent.  —  If  y  is  absent,  the  equation  is 
of  the  form 


dy 
If  we  put  ~  =/,  then 


dp          dny  _ 
"'          ~ 


and  the  equation  to  be  solved  is 

Jdn~lp    d^p        dp          \_ 
J  \£F*  '  dx«-"  "  *'  ~dx>  p>  *)  -  °' 


which  is  of  order   n  —  i  .     If  this  can   be  solved   for  /,  we   have 
p  =  $(x,  flt  c2,  '..,  cn_i),  and  y  can  be  obtained  by  the  quadrature 


132  DIFFERENTIAL   EQUATIONS  §57 

More  generally,  if  y  and  all  of  its  derivatives  up  to  the  (r  —  i)st 
are  absent,  so  that  the  equation  is  of  the  form 


by  letting  —~  =  v,  the  equation  becomes 

~*v         dv 


which  is  of  order  «  —  r.     If  this  can  be  solved   for  v,   we   have 
v=$(x,  Ci,  cz,  •-,  tn-r)>  and  y  can  be  obtained  by  r  successive 

quadratures,  i.e.  y  =  ff  •••    C  <j>(x,  cl9  cz,  ••-,  cn_^do?  +  cn-r+ixr~l 

...  +  c^x  +  cn. 


If  y  and  all  of  its  derivatives  except  the  highest  are  absent, 
equation  may  be  put  in  the  form 


and  the  solution  is  obtained  directly  by  n  successive  quadratures. 
F°r          ~  =     f(X^X  +  ai>  whence  ' 


and  so  on,  until  we  get 


§57  HIGHER  ORDER  THAN  THE  FIRST  133 

dy 
Putting  -4-  =/,  we  have 


.-.  tan"1/  =  c  —  tan"1  x,  or  /  =  -^ — — ,  where  c±  =  tan  c 
Integrating,  we  have  c*y  =  (c-f  +  i)  log  (i  -f  ^x)  —  ^x  +  c2. 

Ex.2,     f  JC-r-5 ^51    =  I  — r-s  I    4-1. 


Putting  -r^v  and  solving  for  this,  we  have 


dv 
— 
dx 


This  is  Clairaut's  form  (§27)  and  has  for  solution 


Integrating,  we  get  =    *2  ± 


Integrating  again,  ^  =  ^  ±  —  V7T~i  +  c'x  +  ^'. 

6  2 


Ex.  3.   -  . 

dx 


Ex...        -^. 


134  DIFFERENTIAL   EQUATIONS  §58 

58.   Independent  Variable  Absent.  —  If  x  is  absent,  by  taking  y  as 

the  independent  variable  and  letting  -j-,=p,\>z  the  dependent  one, 
we  have 


9 

dydf         \dy)  ' 

+7^^+II/ 

dy*  +  1P  dy  df  ^      P  \dy)  df 


and  the  equation  becomes  one  of  order  n  —  i.     If  this  can  be  solved 
for/,  we  have/=  <f>(y,  c^  c^  ••-,  ^«_i),  and  y  can  be  obtained  by  the 

quadrature  I  —  -  -  ^  -  -  =  x  + 
J  <j>(y,  flf  c*  "-,  cn_i) 


c. 


Remark.  —  The  equation  —  %  =  /(/),  which  belongs  to  the  class  of  equations 
here  considered,  has  the  obvious  integrating  factor  2  -2-  dx.      Using  it,  we  have 


dx 


Integrating,  we  get  =2     /(  y}  dy  + 


whence 


It  should  be  noted  that  the  general  method  of  this  paragraph  leads  to  exactly 
this  method  of  solution. 


§59  HIGHER   ORDER  THAN  THE  FIRST  135 


The  factor  p  =  o  gives  y  =  c,  a  particular  solution. 

y  —  —p—f  =o  has  the  obvious  integrating  factor  —^.     Using  this 

we  have   •£  =  y  +  c.    Remembering  that  p  =  -+-,  we  have 
y  dx 

_dy__dx 

—  —  •  —  -  —  ax, 

y(y  +  ') 

whence  log  —2—  —  ex  +  <*  : 

y  +  c 


or 


E,2.  + 

^c2 

Ex.3.    a 


=  0. 


59.  Linear  Equations  with  Particular  Integral  Known.  —  If  the 
equation  is  linear  and  of  any  order,  and  a  particular  integral  is 
known  when  the  right-hand  member  is  made  zero,  the  method  i°, 
§  53>  applies.*  Thus,  let  the  equation  be 

.-  +Pn_lZ>+Pn)y=X. 


*  The  hint  there  given  as  to  how  a  particular  integral  may  at  times  be  found,  applies 
equally  well  here. 


136  DIFFERENTIAL   EQUATIONS  §59 

Putting  y  =y&t  we  have  Dy  =  Dy±  •  v  -\  ----  ,* 


Making  the  substitution,  we  have 


By  hypothesis,  the  coefficient  of  v  is  zero.     Hence,  on  letting  —  =/, 
the  equation  reduces  to  one  of  order  n  —  i  . 

Ex.1.    [(x2-2x+2)ZP-x2D2+2xD-2]y  =  o. 
y  =  x  is  a  particular  solution.     Putting  y  =  xv,  we  have 


Letting  ZPv  =  q,  this  becomes 


X  X*—2X+2 


Integrating, 

and 

Therefore  y  =  c^  -f  c&?  +  £?*• 

*  ...  stands  for  terms  free  of  v,  and  involving  its  derivatives  to  an  order  as  high  as 
the  exponent  of  D  on  the  left. 


§60  HIGHER  ORDER  THAN  THE   FIRST  1  37 

Ex.2. 


By  inspection,  it  is  seen  that  e",  e~x,  x  are  particular  integrals, 
hence  we  know  at  once  that  the  complementary  function  is 

Y  =  c^e  +  <:#-*  +  czx. 

The  student  should  verify  that,  by  the  method  of  variation  of 
parameters  (§  49),  this  becomes  the  general  solution  when 

C\  =  — 
2 


X 

Hence  the  solution  is 


In  order  to  get  practice  in  the  general  method  of  this  paragraph, 
let  the  student  solve  this  example  by  that  method. 

60.  Exact  Equation.  Integrating  Factor.  —  In  case  the  equation 
is  the  derivative  of  another  one,  the  order  may  be  reduced  by  direct 
integration.  No  simple  formula  can  be  given  as  a  test  for  exactness 
(except  in  the  case  of  linear  equations).  But  the  method  is  simple 
and  direct,  and  can  probably  be  brought  out  best  by  the  following 
examples  : 

Consider  first  the  linear  equation, 


138  DIFFERENTIAL  EQUATIONS  §60 

PO  —  will  arise  on  differentiating  P0-~.     But  differentiating  this, 
do<?  dor 

we  get  PQ  —  ^  +P'o  —  ~)  (indicating  differentiation  by  a  prime).     Now, 

itOC^  dOC' 

if  (i)  is  exact,  so  is 


(/>  _  />  ')    £  will  arise  on  differentiating  (/>,  -/>'<,)      .     But  differ- 

(IOC  CtOC 

entiating  this,  we  get  (P,  -P'0)  ^2  +  (P\-P"0)  &  . 

dor  ax 


Hence  if  (2)  is  exact,  so  also  is 

(3)  (f,-f 


(P2-P\-\-P"Q)^  will  arise  on  differentiating  (/>2-/>'1  +  />"0)> 
dx 

But  differentiating  this,  we  get 

(/>,  -  P\  +  P"0)  &  +  (/»,  -  ^  +  />'»0)  j, 
^ 

hence,  if  (3)  is  exact,  we  must  have  Ps  —  P'.2  +  P"l  —  P'"0  =  o.* 

Moreover,  this  condition  is  also  obviously  sufficient,  and  we  have 
that  a  first  integral  of  (i)  is 


*  This  suggests  the  condition  for  exactness  of  a  linear  equation  of  the  «th  order, 

(4)  Pn  -P'n-l 


§60  HIGHER  ORDER  THAN  THE   FIRST  139 

This  method  applies  also  to  equations  that  are  not  linear,  but  in 
such  cases  there  is  no  simple  test  for  exactness  ;  one  must  actually 
carry  out  the  work  of  finding  the  first  integral  to  find  out  whether  it 
is  exact.  Thus  consider  the  equation 


dx*  dx  d 

The   derivative  of        +  *          is 


Subtracting  this,  we  have  4  y  + 

dx  dx2        \dx 

This  is  the  derivative  of  2  y  I  -2-  J  .     Hence,  a  first  integral  is 

\dxj 


Let  the  student  show  that  this  is  also  exact. 

Remark.  —  Since  an  exact  differential  results  from  differentiating  an  expression 
of  one  lower  order,  it  is  obviously  necessary  that  in  it  the  highest  ordered  derivative 
appear  to  the  first  degree  only.  In  other  words,  we  can  never  expect  an  expres 
sion,  in  which  the  highest  derivative  entering  appears  to  a  higher  degree  than  the 
first,  to  be  exact.  Moreover,  this  must  be  true  of  all  the  expressions  [such  as  (2) 
and  (3)  above]  which  arise  in  the  course  of  the  process.  If  any  one  of  these 
turns  out  to  be  of  higher  degree  than  the  first,  there  is  no  need  to,  proceed  farther. 


This  is  linear  and  satisfies  the  condition  (4)  for  exactness. 
The  derivative  of  O+2)2         is(*+  *Y        +  2  (x  +  2) 


Subtracting,  we  get  -  (x  +  2)        +      • 


I4O  DIFFERENTIAL   EQUATIONS  §60 

The  derivative  of  -  (x  +  2)  ^  is  -  (x  -f  2)  ^  -  ^  .     Subtract- 

2 

-^ 


ing,  we  get  2-^,  whose  integral  is  27.     Hence,  a  first  integral  is 


Since  2  -f  i  +  2  =£  o,  this  equation  is  not  exact. 
But  putting  #4-  2  =  **  (§  51),  we  have 


a  linear  equation  with  constant  coefficients. 
The  roots  of  the  auxiliary  equation  are  i  ±  /. 
Hence  Y=  f  (A  cos  z  +  B  sin  z)  is  the  complementary  function. 

For  the  particular    integral    try   U—af  +  b.       Substituting   in    the 

ci 
equation,  we  must  have  aez  -\-  2  b  =  e*  -{-  c'.     Hence  a  =  i,  b  =  —  • 

And  the  solution  is  y  =  e*  (A  cos  z  +  B  sin  z)  +ez  +  -  >  or 


y  =  (x  +  2)  \_A  cos  log  (#  +  2)  -f  #  sin  log  (x  +  2)  ]  +  x  -f  C. 

Remark.  —  It  may  be  noted  that  since  y  is  absent  in  the  original  differential 
equation,  the  method  of  §  57  applies.  The  student  should  solve  the  problem 
from  this  point  of  view. 


Ex.3.     a:_I' 


§60  HIGHER  ORDER  THAN  1HE  FIRST  14! 

Ex.4.     *_, 


Ex.5. 

do  dx     o  do 

-- 


o. 


It  is  at  times  possible  to  find  an  integrating  factor.  But  no  general 
treatment  of  this  part  of  the  subject  can  be  given  here.*  In  the 
cases  to  be  considered  here,  special  methods,  or  inspection,  will  be 
employed.  One  important  type  of  equation,  arising  in  physical 
problems,  has  already  been  mentioned  (Remark,  §  58). 


Ex.6. 


This  equation  is  not  exact,  since  —  2#3+i  — 
But  xm  will  be  an  integrating  factor  provided  we  can  find  a  value 
for  m  such  that 


-  2  xm+3  +  xm  -  2  ( 


or  (w2  +  7  m  +  10)  xm+3  +  (m  +  2)  xm  =  o  ; 

i.e.  w2  +  7  w  +  10  =  o,  and  w  +  2  =  o. 

Both  of  these  will  be  satisfied  if  m  =  —  2. 


*  For  integrating  factors  in  the  case  of  linear  equations,  see  Schlesinger,  Different 
tialgleichungen^  p.  147,  and  references  given  there. 


142  DIFFERENTIAL  EQUATIONS  §60 

Hence  x~2  is  an  integrating  factor.     Using  it,  we  have 


ax-  ax 


dx 


-(2  *- 


Hence  a  first  integral  is  x3-^-  —  (x2  +  x~l)y  =  c,  or 


dx 
This  is  linear.     An  integrating  factor  is 

-/(X-1+X-4X*   _        -1 0*-3          (§     13). 
C       J  — —  »^v        C»  \°         O/ 

.'.^-^a*-8  =  <:  r^-V^x-3rfa;4- c'=—ce$x~*+  c',  or 


y  -\-  cx 

Ex.7.    x*(i-x*)£>2y-x3Z)y-2y  =  o. 

Ex.  8.    x*ZPy  —  5  x  D^y  +(4^  +  5)^—8  x*y  =  o. 

Ex.9.    g  +  /(,)£ 
dx*  dx 

*  This  type  of  equation  was  first  treated  by  Joseph  Liouville  (1809-1882).  Let  the 
student  show  that  it  is  the  differential  equation  corresponding  to  a  primitive  of  the  form 
ff  (y)  =  a$(x)  +  b,  where  F  and  *  are  any  functions  of  their  respective  variables,  and  a 
and  b  are  the  arbitrary  constants  to  be  eliminated. 


§61  HIGHER  ORDER  THAN  THE  FIRST  143 

By  inspection  f -2j   -is  seen  to  be  an  integrating  factor. 

Introducing  this,  we  have 


whence        log  fj£\  +  J/  (x)  dx  +  f  4>(y)dy 


or 


and 


£e 


Ex.10.          +2cot*       +  2tan 

oar  dx  dx 


61.  Transformation  of  Variables.  —  In  case  the  equation  to  be 
integrated  does  not  come  under  any  of  the  heads  already  treated,  it 
is  possible,  at  times,  to  reduce  it  to  one  of  them  by  a  transformation. 
No  general  rule  for  this  can  be  formulated.  The  form  of  the  equa 
tion  will  frequently  suggest  the  transformation  to  be  tried. 


Ex.  1. 

dx 

The  set  of  terms  (x^-  —  y\    suggests  the  transformation  y  —  vx. 
\    ax       J 

Making  this  transformation,  the  equation  becomes  (after  dropping  the 
factor  x*\ 


dx 


144  DIFFERENTIAL   EQUATIONS  §  62 

This  is  exact,  and  has  for  first  integral 

dx      2 

This  is  also  exact,  giving  xv*  =  c^x  -f-  c2t 

or  /  =  c^  -f-  c&. 

[A  less  obvious  transformation  is  jy2  =  v.     Let  the  student  solve 
the  problem  by  making  this  transformation.] 

Ex.2.   gtU-fcar-yf-O. 

dx 


Ex.3.  ,_=/k)g,-.*y.     [Let  logj—f-.Le.  >=«•. 


Ex.4,    sin2.*  —  ^~  —  2j=o.     [Let  cot  #  =  £.] 

If  the  more  obvious  transformation  sin  x  =  z  is  made,  the  resulting 
equation  can  be  made  exact  by  multiplying  by  a  proper  power  of  z, 
and  can  then  be  integrated. 

62.  Summary.  —  The  number  of  classes  of  differential  equations 
of  higher  order  than  the  first  for  which  a  general  method  of  solution 
is  known  is  very  small.  We  can  tell  by  inspection 

i°  when  the  dependent  variable  is  absent  ;  let  the  lowest  ordered 
derivative*  that  appears  be  a  new  variable  (§  57); 

2°  when  the  independent  variable  is  absent  ;  let  the  first  deriva 
tive  of  the  dependent  variable  be  a  new  variable,  and  consider  the 

*  Provided  this  is  not  also  the  highest  ordered  derivative  that  appears.  If  such  is 
the  case,  let  the  next  lower  ordered  derivative  be  a  new  variable. 


§62  HIGHER   ORDER  THAN    THE   FIRST  145 

dependent  variable  as  the  independent  one  (§  58)  ;  in  particular,  if 
the  equation  has  the  form  — ^  =/(y),  this  method  leads  to  the  ob 
vious  integrating  factor  -2-  dx  (§  58,  Remark). 
doc 

3°  If  the  equation  is  linear,  and  a  particular  integral  y±  can  be 
found  when  the  right-hand  member  is  made  zero,  let  y  =  y\v,  and 

in  the  transformed  equation  put  —  =/  (§  59). 

ClOC 

4°  If  the  equation  is  linear  and  of  the  second  order,  the  methods 
of  Chapter  VIII  may  apply  (§  55). 

If  none  of  the  above  cases  occur,  test  the  equation  for  exactness 
(§  60).  Should  this  not  prove  to  be  the  case,  some  special  device 
must  be  resorted  to,  such  as  finding  an  integrating  factor  (§  60),  or 
finding  some  suitable  transformation  (§  61). 

As  a  final  resort,  the  method  of  integrating  in  series  may  be  tried 
(§  74)- 


Ex.4.   (I+^)liZ  +  g^^+i8*^  +  6.>>  =  o. 
Ex.  5.   (#*- 
Ex.  6.    v(i- 


146  DIFFERENTIAL  EQUATIONS  §6* 

Ex.    7. 


Ex.    8.    W*+2y)^+2*m  +4(*+>^4-2y  +  .*2==o. 


EX.  9. 


Ex.10.     «_ 


Ex.11. 


Ex.12,   sin^-^  — CQSJC-^  +  2  sin^c  •  v  =  o. 


Ex.  13.  Determine  the  curves  in  which  the  radius  of  curvature  is 
equal  to  the  normal,  (a)  when  the  two  have  the  same  direction, 
(£)  when  they  have  opposite  directions. 


The  radius  of  curvature  =  —  *=  —  -^  —  —  =*  •      The    normal    being 


supposed  drawn  toward  the  axis  of  x,  when    it  and  the  radius  of 


curvature  are  drawn  in  the  same  direction,  y  and   — —0   have  opposite 

ax1 

signs ;  and  when  drawn  in  opposite  directions,  y  and  — £  have  the 


§62  HIGHER  ORDER  THAN  THE   FIRST  147 

Ex.  14.  Determine  the  curves  in  which  the  radius  of  curvature  is 
twice  the  normal,  (a)  when  the  two  have  the  same  direction,  (#) 
when  they  have  opposite  directions. 

Ex.  15.  Find  the  curves  whose  radius  of  curvature  is  k  times  the 
cube  of  the  normal. 

Ex.  16.  A  particle  which  sets  off  from  a  point  of  the  axis  of  x,  at 
a  distance  a  from  the  origin,  moves  uniformly  in  a  direction  parallel 
to  the  axis  of  y.  It  is  pursued  by  a  particle  which  sets  off  at  the 
same  time  from  the  origin,  and  travels  with  a  velocity  which  is  n  times 
that  of  the  former.  Required  the  path  of  the  latter. 

[This  path  is  usually  referred  to  as  the  curve  of  pursuit.  Its 
differential  equation  may  be  obtained  from  the  following  considera 
tions  :  Let  (x,  y)  be  the  coordinates  of  the  pursuing  point,  (£,  rj)  those 
of  the  point  pursued.  The  path  of  the  latter  being  known,  we  have 
given  (i)/(£,  if)  =  o.  Since  the  point  pursued  is  always  in  the  tan 

gent  to  the  curve  of  pursuit,  we  have  (2)  rj  —y  =  ^-(g  —  x).     (i)  and 

j  ttOC 

(2)  determine  £  and  77  in  terms  of  x,y,-^-.     If  the  velocities  of  the 
point  pursued  and  pursuing  point  are  as  i  :  n,  we  have 


or  taking  x  as  the  independent  variable, 


Substituting  in  this  the  values  of  £  and  rj  from  (i)  and  (2),  we  obtain 
the  differential  equation  of  the  curve  of  pursuit.] 

Ex.  17.  Find  the  velocity  of  the  weighted  end  of  a  simple  pen 
dulum  of  length  /,  swinging  in  a  vacuum,  if  at  the  time  /=o,  v=  o 
and  0  =  a,  where  «  is  not  so  small  that  sin  a  may  be  replaced  by 
a  as  a  first  approximation.  (See  Ex.  17,  §  52.) 


148  DIFFERENTIAL   EQUATIONS  §62 

Ex.  18.   A  particle  moves  in  a  straight  line  attracted  by  a  force 
varying  inversely  as  the  square  of  the  distance.     [Equation  of  motion 

dzx  &  ~1 

is  — -  =  —  — .        If  it  starts  with  zero  velocity  at  a  distance  a  from 

the  center  of  the  force, 

(a)    find  its  velocity  at  any  point  in  its  path, 
(£)    find  the  time  required  to  reach  that  point, 

(c)  how  far  will  it  have  to  move  in  order  to  acquire  the  same 
velocity  with  which  it  would  arrive  at  the  point  a  if  it  had  started 
to  move  from  infinity  with  zero  initial  velocity. 

(d)  Since  gravity  acts  according  to  the  above  law,  find  the  velocity 
with  which  a  body  (a  meteorite,  for  example)  will  strike  the  surface 
of  the  earth  if  it  falls  from  a  distance  h  above  the  surface. 

[Acceleration  due  to  gravity  at  the  earth's  surface  is  usually  des 
ignated  by^.     Hence  k*  =  gRz,  if  R  is  the  radius  of  the  earth.] 


CHAPTER  X 
SYSTEMS  OF   SIMULTANEOUS  EQUATIONS 

63.  General  Method  of  Solution.  —  It  is  proved  in  the  general 
theory  of  ordinary  differential  equations  that  a  system  of  n  equa 
tions  involving  n  dependent  variables  can,  in  general,  be  solved 

(§  70). 

We  shall  consider  here  the  case  of  n  —  2,  the  method  admitting  of 
being  extended  to  any  number.  Let  the  equations  be 


(0 

(2)  /2[(Xw  (y),,  /]  = 


where  the  highest  ordered  derivatives  of  x  appearing  in  (i)  and  (2) 
are  respectively  m  and  m  +/,  those  of  y  are  r  and  s,  and  /  is  the 
independent  variable.     Of  course  /  may  be  zero. 
Differentiating  (i)  /  times,  we  get  successively 

(3) 
(4) 

(P  +  2)  /P+2  [(*)»+,,,  (y\+P,  /]  =  O. 

We  now  have/-f  2  equations  from  which  to  eliminate  x  and  all  of 
its  m  -\-p  derivatives.  In  general  (unless  m  =  o)  this  will  not  be 
sufficient,  for  we  must  have  one  more  equation  at  our  disposal  than 
the  number  of  quantities  to  bf  eliminated.  We  proceed  now  to 


150  DIFFERENTIAL   EQUATIONS  §64 

differentiate  both  (2)  and  (p  -f-  2).  Since  this  introduces  two  new 
equations  and  only  one  new  derivative  of  x,  we  see  that,  by  repeating 
the  process  the  proper  number  of  times,  the  number  of  equations  will 
exceed  that  of  the  quantities  to  be  eliminated  by  unity.  Performing 
the  elimination,  we  have  a  single  equation  in  y.  Integrating  this 
and  substituting  the  value  of  y  in  (i),  we  have  an  equation  in  x 
only,  which  must  then  be  solved. 

Remark.  —  It  is  almost  needless  to  add  that  we  may  first  eliminate  y  and  its 
derivatives,  and  then  solve  for  x. 

Or,  we  may  solve  for  x  and  for  y  separately.  In  this  case  the  constants  of 
integration  arising  are  not  all  independent.  The  relations  among  them  can  be 
found  by  substituting  in  one  of  the  equations  (i)  and  (2). 

64.  Systems  of  Linear  Equations  with  Constant  Coefficients.  —  This 
method  can  be  carried  out  very  readily  in  case  the  equations  are 
linear  and  the  coefficients  constants.  Thus  consider  the  example 


These  may  be  written 

(1)  (D  +  i)  x  —  Dy  —  cos  f, 

(2)  (£>2  +  3)* 

Differentiating  (i),  we  have 

(3)  (D*  +  D)x 

We  must  eliminate  x,  £>x,  LPx  •  this  requires  four  equations. 
Hence  we  must  differentiate  (2)  and  (3).     This  gives  rise  to 

(4)  (l?  +  SD)x-(&  +  D)y=2<!*9 

(5)  (J?  +  D*)x-l?y=-~  cos  /. 


§64  SYSTEMS   OF  SIMULTANEOUS   EQUATIONS  151 

We  have  now  five  equations  from  which  we  can  eliminate  the  foui 
quantities  x,  Dx,  Dzx,  £Px.  By  taking  —  3  X  (i),  i  x  (2),  i  x  (4), 
—  i  X  (5),  and  adding,  we  get 


(6)  (I?-ir  +  D-i)y  =  se*t-2  cos/, 

which  is  a  linear  equation  in  y  only,  and  can  be  solved  readily. 

Before  doing  so,  however,  we  shall  see  how  (6)  can  be  gotten 
directly  from  (i)  and  (2).  Since,  when  looked  upon  algebraically, 
(3)  and  (5)  are  respectively  D  and  Z>2  times  (i),  and  (4)  is  D  times 
(2)  (temporarily  supposing  their  right-hand  members  to  be  zero), 
the  above  method  of  elimination  amounted  to  subtracting  (D~  +  3) 
times  (i)  from  (D  +  i)  times  (2).  But  this  is  precisely  the  method 
we  would  have  pursued  in  eliminating  x  from  (i)  and  (2)  had  D  and 
its  powers  been  algebraic  quantities  instead  of  operators.  Now,  so 
long  as  the  equations  are  linear  with  constant  coefficients,  this  process 
is  always  allowable,  since  it  involves  only  the  operations  of  addition, 
subtraction,  and  multiplication  with  the  operator  D.  Hence  we  need 
only  write  our  equations  in  the  form  of  (i)  and  (2),  solve  them  as 
algebraic  equations,  remembering,  however,  that  D  is  an  operator  in 
case  there  are  any  terms  in  the  right-hand  members.  In  practice  it 
is  frequently  convenient  to  use  determinants.  Thus  solving  (i)  and 
(2)  for  y,  we  have 

D   + 1      —     D  _    Z>  +  i     cos  / 

or 

(6)  (D*  —  LP  +  D  —  i)y  =  3  <?2'  —  2  cos  /. 


The  complementary  function  is  Y—  c^  -f  cz  sin  /  +  ^3  cos  /. 
For  the  particular  integral,  try  U=  ae2t  +  ^/sin  /+  ct  cos  /. 

Substituting  this  for  y  in  (6),  we  find  that  «  =  ^,    b=-,    <:  =  - 

5  2  2 

(7)     .-.  ^  =  ^  +  <r2sin/+^3Cosf+3<?*  +  -/sin/4-- 

2  2 


152  DIFFERENTIAL   EQUATIONS  §64 

To  find  x  we  may  substitute  this  value  in  either  (i)  or  (2),  and 
solve  the  resulting  equation  in  x* 

Or  we  can  treat  x  exactly  as  we  did  y,  that  is,  solve  (i)  and  (2) 
directly  for  x.  Doing  this  we  have 


D  +  i      -       D 


X  = 


cos  /     —      D 

*          -  (/>+!) 


D     cos/ 
D  +  i       e* 


or 

(8)  (Z>3-  &  +  D-  i)  x=  2  e*  +  sin  /-  cos  /.f 

Solving  this,  we  have 

(9)  x  =  c^  <?<  +  cj  sin  /  +  s3'  cos  /  +  -<?2<  +  -/cos  /. 

But  these  constants  are  not  independent  of  those  in  (7).  They 
may  be  found  by  substituting  (7)  and  (9)  in  either  of  the  original 

equations  and  equating  coefficients.     Doing  this,  we  find  ^1'  =  -c1) 

*'=afo-r8)  +  3,  r3'=i(V2  +  <3)+-- 
2  42  4 

*  In  general,  in  solving  for  the  variable  first  eliminated,  it  is  necessary  to  solve  a 
differential  equation.  For  example,  if  we  put  the  value  of  y  given  by  (7)  in  (i),  we 
have  an  equation  of  the  first  order  to  solve ;  if  we  put  it  in  (2) ,  we  have  an  equation  of 
the  second  order  to  solve.  The  new  constants  of  integration  that  arise  now  are  not 
arbitrary,  but  must  be  determined  so  that  the  other  equation  is  also  satisfied.  This  is 
done  by  substituting  in  the  other  equation,  and  equating  coefficients.  In  this  particu 
lar  example  it  would  have  been  simpler  to  have  solved  for  x  first.  The  value  of  y  could 
then  be  gotten  immediately  from  the  equation  resulting  from  subtracting  (i)  from  (2). 
Let  the  student  do  this. 

t  We  see  by  this  method  of  solution  that  the  differential  equations  in  x  and  in  y, 
each  resulting  from  the  elimination  of  the  other  variable,  have  the  same  left-hand 
members,  and  that  the  complementary  functions  are  therefore  of  the  same  form  in  the 
case  of  the  two  variables.  This  is  obviously  true  in  the  case  of  n  dependent  variables 
defined  by  «  linear  equations  with  constant  coefficients. 


§65  SYSTEMS   OF   SIMULTANEOUS   EQUATIONS  1 53 

Hence  the  general  solution  of  our  system  of  equations  is 
4  x  =  2  c^  +  (2  ^2  —  2  c&  4-  3)  sin  /  4-  (2  <r2  4-  2  <r3  4-  i)  cos  / 

8 
5 

y  =  f^  4-  <r2  sin  /4-  <r3  cos  /  4-  ^«?2<  4-  -  /  (sin  t+  cos  t). 


f3^ 


Ex.  1. 


Ex.2. 


dt 


Ex.  3. 


€^.  Systems  of  Equations  of  the  First  Order.  —  If  the  equations 
are  of  the  first  order,  we  can  suppose  them  solved  for  the  first  deriva 
tives  of  each  of  the  dependent  variables.  [We  shall  consider  the 
case  of  two  dependent  variables.  But  the  methods  here  brought  out 
obviously  apply  to  the  case  of  n  such  variables.]  Let  the  system  be 


154  DIFFERENTIAL   EQUATIONS  §65 

The  general  method  of  §  63   applies.     But  in  certain  cases  the 
solution  can  be  brought  about  very  much  more  readily.     It  is  some 
of  these  cases  that  we  shall  consider  now.     (i)  can  be  written  in  the 
more  symmetrical  form 
(2\  dx^dy_dff 

P      Q      R 

i°  One  of  these  equations  may  involve  only  two  of  the  variables, 
or  it  may  be  possible,  by  a  proper  choice  of  a  pair  of  members  of 
(2),  to  strike  out  a  common  factor  so  as  to  obtain  an  equation 
involving  only  two  of  the  variables.  Thus,  to  fix  the  ideas,  suppose 

that  /  does  not  appear  in  —  —  ^  ,  or  can  be  removed  from  it.     We 
have,  on  solving  this, 

(3)  4>(x,y)  =  £i- 

If  this  can  be  done  a  second  time,  so  that  a  second  relation  of  the 
form 

(4)  *(y,  0  =  ^ 

can  be  found,  then  the  complete  solution  consists  of  the  two  rela 
tions  (3)  and  (4). 


Ex.1.         =      =      . 

yt      tx      xy 

From  the  first  two  members  we  have  x2  —  y'*  =  tl.  1 
From  the  last  two   members  we  have  y*  —  t2  =•  cz.  \ 
(Using  the  first  and  last  members  we  get  f-  —  x~  =  c.     But  this  is 
obviously  not  distinct  from  the  other  two.) 

2°  If  we  can  find  only  one  integral  expression  of  the  above  type, 
say  (3),  we  can,  by  means  of  it,  express  one  of  the  variables  in  terms 
of  clt  and  the  other ;  thus,  to  fix  the  ideas,  we  can  solve  (3)  for  x  in 

terms  of  ^  and  y.     Substituting  this  value  of  x  in  -^  =  — ,  we  have  an 

Q    & 


§65  SYSTEMS   OF   SIMULTANEOUS   EQUATIONS  1  55 

equation  involving  jy,  /,  and  the  constant  c^     Solving  this,  we  have  a 
second  relation 

(5) 


(3)  and  (5)  together  constitute  the  general  solution. 
At  times  it  is  desirable  to  replace  c\  in  (5)  by  its  value  in  terms  of 
x  and  y.     The  solution  is  then 


Ex.2.          =      =      . 

xt     yt      xy 

TT  ,  dx      dv      ,  x 

Here  we  have         —  =  -^-,  whence  — 

x       y  y 


.*.  x  =  c\y  ',  and  we  have 

•  =  ~>  or 


whence  c^f~  —  /2  =  c2t 

or  jcy  —  /2  =  <r2. 

/.  The  solution  is  {  *  ~  ^y  =  0> 

l^y-/2  =  ^2. 

3°  It  sometimes  happens  that  we  can  find  multipliers  \(x,  y,  /), 
lf.(x,yt  /),  v  (x,y,  t)  such  that,  making  use  of  the  fact  that 

dx  _  dy  _dt  _  \dx  +  /x  dy  +  vdt 
P      Q  ~R~  X/>+/x<2  +  vR  ' 


(a)  this  last  member  when  combined  with  one  of  the  others  gives 
rise  to  an  equation  which  can  be  solved  ;  or 


156  DIFFERENTIAL  EQUATIONS  §65 

(6)  XP  +  pQ  +  vR   may   equal    zero,    at    the    same    time    that 
X  dx  +  fM  dy  +  v  dt  =  o    satisfies    the    condition    for    integrability 

(§  35)  ;  or 

(c]  by  a  choice  of  two  sets  of  multipliers, 

X1  dx  +  fa  dy  -f  vi  dt  _  X2  dx  4-  /x2  ^  + 


may  be  solvable. 

If  we  can  find  two  independent  relations  by  any  of  these  methods, 
each  involving  an  arbitrary  constant,  we  have  the  general  solution. 


EX  3       =    = 

y       x        t 

From  —  =  -^  we  have  a?  —  y* 

y       x 


Letting  A.  =  //,  =  i,  v  =  o,  we  have 

dx  -4-  dy      dt     , 

-  1  —  £.  =  —  whence  x  +  y  =  cJ. 

x+y        t 


Ex.4.        dx  dv  dt 


cy  —  bt      at  —  ex      bx  —  ay 
Letting  A.  =  a,  /u,  =  b,  v  =  c,  we  have  by  composition  that  the  com- 
mon  ratio  is  equal  to  g*-±* <»+"*. 

O 

.'.  a  dx  +  b  dy  -f  c  dt  =  o,  whence  ax  +  by  +  ct  =  c±. 

Similarly,    letting  X  =  x,  p  =  y,  v  =  /,  we  have 

x  dx  +  y  dy  +  tdt  =  o,  whence  x2  +  J2  +  ^2  =  ^2- 


Ex.  5. 

2xy 


§66  SYSTEMS   OF   SIMULTANEOUS   EQUATIONS  1 57 

Letting  Xl  =  i,  ^  =  i,  v±  =  o,  and  X2  =  i,  /x2  =  —  i,  v2  =  o,  we   have 

/T^V    \    /7'V        doc (L V  I  I 

(*+7)2      (x—  y)2'  x+y     x—y 

Asain    7^r4* = ,..?'*<  > whence  *  +y  =  ^« 


Ex.    6.  =         =     . 


dx  __dy  _     dt 

~  —  - 

-  X          1  + 


Ex.    8.   !     =      = 
Ex.    9. 


yt      xt      x  +y 

dx  dv         dt 


_    yl     _ 


Ex.10.     ^          dv  dt 


Ex.  11. 
Ex.  12. 


t      t  +  x      x+y 

dx  dv  dt 


•^    i  ^ 

^r  dy 


9  /(.r5  — 


66.    Geometrical  Interpretation.  —  P(x, y,  z)*  Q(x,  y,  z),  R(x,  y,  z) 

may  be  looked  upon  as  determining  the  line =  — — -^  = 

through  the  point  (x,  y,  z),  which  is  any  point  in  space.     An  integral 
curve  of  the  system  —  =  -^  =  -'- z  will  be,  then,  a  curve  such  that,  at 

*  From  now  on  we  shall  use  the  three  letters  x,y,  z,  instead  of  x,y,  t. 


158  DIFFERENTIAL   EQUATIONS  §66 

each  point  of  it,  it  is  tangent  to  the  line  at  that  point  determined  by 
P,  Q,  R.  The  general  solution,  we  have  seen,  consists  of  two  rela- 

{  u  (x,  y,  z]  =  c\  } 

tions,    i      :  '          ^  ,  involving  two  arbitrary  constants.     That  is, 

I  v  (x, }',  z)  —  c2 ) 

the  general  solution  represents  a  doubly  infinite  system  of  curves, 
which  are  the  intersections  of  two  singly  infinite  systems  of  surfaces.* 
Thus  in  the  case  of  Ex.  3,  §  65,  the  integral  curves  are  the  intersec 
tions  of  the  family  of  cylinders  x~  —y-  —  ^  with  the  family  of  planes 
x  +  y  —  f-2Z=  o.  In  §  40  we  saw  that  a  solution  of  the  total 
differential  equation  P  dx  +  Qdy  +  Rdz=o  represents  a  surface 
such  that,  at  each  point  (x,  y,  z)  of  it,  it  is  tangent  to  the  plane 
P(X—  x)+  Q(Y  —  y)+jR(Z—z)=o,  the  direction  cosines  of  whose 
normal  are  proportional  to  Pt  Q,  R.  Hence  we  see  that  the  inte 
gral  curves  of  —  =  -2-  =  —  cut  orthogonally  any  integral  surface  of 
P  Q  R 

Pdx -\-  Qdy  -f-  Rdz  =  o.  Since  we  can  find  a  family  of  integral  sur 
faces  of  Pdx  -\-  Qdy  +  R dz  =  o  only  when  Pt  Q,  R  satisfy  the  con 
dition  for  integrability  [§  35,  (3)],  we  see  that  only  in  this  case  will 
there  exist  a  family  of  surfaces  of  which  the  family  of  integral  curves 

of  —  =  -^  =  —  are  orthogonal  trajectories.    Thus,  since yz  dx  -\-zxdy 
P      Q      R 

+  xydz=o  has  xyz=c  as  its  general  solution,  we  see  from  Ex.  i, 
§  65,  that  the  curves  of  intersection  of  the  cylinders  x2  —  y2  =  c±  and 
y2  —  z2  =  C<L  are  cut  orthogonally  by  the  family  of  surfaces  xyz  =  c. 
On  the  other  hand,  since  xzdx +yzdy -\- xydz  =  o  does  net  satisfy 
*  Supposing  P,  Q,  R  single-valued  functions  of  x,  y,  z,  there  passes  through  any 
point  (x0,y0,  «0)  the  single  curve  f  *J*'*  *|~*/^**0l*°|  }•  since  the  differential 
equations  determine  a  single  direction  at  each  point  in  space.  If  P,  Q,  /?  are  not  all 
single-valued  functions,  that  is,  if  the  differential  equations  are  not  both  of  the  first 
degree,  then  more  than  one  line  (or  direction)  will  correspond  to  a  set  of  values  of 
(jc,y,  z),  and  there  will  be  more  than  one  integral  curve  passing  through  a  point.  In 
this  case,  »  and  v  will  not  be  single-valued,  that  is,  when  the  solutions  are  cleared  of 
fractions  and  rationalized,  the  constants  of  integration  do  not  enter  to  the  first  degree. 
This  is  analogous  to  what  we  found  in  the  case  of  a  single  equation  of  the  first  order  in 
two  variables  (§  ao). 


§67  SYSTEMS   OF   SIMULTANEOUS    EQUATIONS  159 

trie  condition  for  integrability,  there  is  no  family  of  surfaces  which  is 
cut  orthogonally  by  the  curves  whose  equations  are  j  2 

(Ex.  2,  §  65).  The  converse  problem  of  finding  the  orthogonal  tra 
jectories  of  a  family  of  surfaces  whose  equation  is  f(x,y,  z)=  t  is 
always  possible,  at  least  theoretically.  For  this  necessitates  solving 
the  system 

dx  _  dy  _  dz 

dx      dy      dz 

Ex.    Find  the   orthogonal  trajectories    of    the  family  of  surfaces 

xy  =  cz. 

67.    Systems  of  Total  Differential  Equations.  —  If  we  have  two 
total  differential  equations  in  three  variables,* 

\dx-\-  Q\dy  +  R^ dz  =  o, 


1  ^dx  +  Q*dy  +  R>>dz  =  o, 

it  can  be  proved  (but  the  limits  of  this  book  will  not  permit  our  doing 
so  here),  that  the  general  solution  consists  of  two  relations  among  the 
variables,  involving  two  arbitrary  constants.  In  actual  practice  we 
proceed  as  follows  : 

If  each  of  equations  (i)  separately  satisfies  the  condition  for  inte 
grability  [§  35,  (3)],  we  solve  each  one,  and  thus  obtain  the  solution 
of  our  system. 

If  only  one  of  the  equations  satisfies  the  condition  for  integrability, 
we  integrate  that  one,  obtaining  a  relation  <f>(x,  y,  z)  =  c±.  Solving 
this  for  one  of  the  variables,  we  replace  this  and  its  derivative  in  the 
other  equation  by  their  values,  thus  giving  rise  to  an  equation  in  two 
varia.bles  only.  Its  solution,  together  with  <j>(x,  y,  z)=cl}  already 
found,  constitutes  the  solution  of  the  system  of  equations,, 

*  The  substance  of  this  paragraph  is  at  once  applicable  to  the  case  of  n  equations  in 
K  4- 1  variables. 


i6o 


DIFFERENTIAL   EQUATIONS 


§68 


If  neither  of  the  equations  is  separately  integrable,  it  is  sometimes 
desirable  to  put  (i)  in  the  form 

f  \  dx  _  dy  _  dz 


where     P=  Q^2  -  <2A  Q  =  RVP2  -  R*P^  R  =  P1Q2- 

The  methods  of  §  65  may  now  be  tried.  If  they  do  not  work, 
then  taking  one  of  the  variables  as  the  independent  one,  say  zt  the 
equations  may  be  written 


(3) 


dz  ""/?' 

dy_Q 
dz~  R 


The  general  method  of  §  63  applies  here. 

68.  Differential  Equations  of  Higher  Order  than  the  First  reducible  to 
Systems  of  Equations  of  the  First  Order.  —  Given  a  single  equation  with  one 
dependent  variable.  We  may  suppose  it  solved  for  the  highest  ordered  derivative; 
thus,  suppose  we  have  the  equation 


(I) 


If  we  put 


dy 


(i)  may  be  replaced  by  the  system  of  three  equations  of  the  first  order 


(2) 


dy 
dx 

dx 


As  an  illustration, 


§69  SYSTEMS  OF  SIMULTANEOUS   EQUATIONS 

is  equivalent  to  the  system 


dx 

~dx 


dx dy dy\ 


Using  the  last  two  terms,  we  have  y2  +yi2  =  fi2t 
or  yi  =  Vci2  —y2. 


whence 


i/ 

=  dx,  or  sin*1  -2-  —x  +  £%t 
c\ 


=  ci  sin  (x  -f  r2). 


In  an  entirely  analogous  manner,  a  system  of  n  equations  of  any  order  in  n 
dependent  variables  may  be  replaced  by  a  system  of  equations  of  the  first  order 
by  letting  each  of  the  derivatives  of  the  dependent  variables  up  to  the  next  to 
the  highest  ordered,  in  the  case  of  each  variable,  be  a  new  variable.  Thus,  by 

the  system  of  equations  of  §  64  may  be  written 


letting  —  = 
dt 


dx 


JL  =  Xi  +  X  —  COS  /, 

dt 


dt 


-  cos  /, 


or 


dy 


x\      x\  +  x  —  cos  t     x\—  2x+  y  •}-  e*  —  cos t 


69.  Summary.  —  To  solve  a  system  of  n  ordinary  differential  equa 
tions  involving  n  dependent  variables  we  differentiate  these  equations 
a  sufficient  number  of  times  to  enable  us  to  eliminate  n  —  i  of  the 
dependent  variables  and  all  their  derivatives,  thus  giving  rise  to  a 
single  equation  involving  only  the  remaining  dependent  variable. 
We  integrate  this,  and  substituting  for  this  variable  and  its  derivatives 
their  values  in  terms  of  the  independent  variable  in  any  n  —  i  of  the 


1 62  DIFFERENTIAL   EQUATIONS  §69 

equations,  we  have  a  new  system  of  n  —  i  equations  in  n  — •  i  depend 
ent  variables.  Repeating  this,  we  find  the  value  of  a  second  variable 
and  reduce  the  number  of  equations  again,  and  so  on.  Or  we  can 
treat  all  of  the  dependent  variables  symmetrically  by  solving  for  each 
one  separately,  and  then  finding  the  relations  among  the  constants 
of  integration  by  substituting  in  some  one  of  the  original  equations 

(§  63)- 

While  this  method  is  frequently  not  practicable,  it  can  be  carried 
out  very  readily  in  case  the  equations  are  linear  with  constant 
coefficients  (§  64). 

If  the  equations  are  of  the  first  order,  special  methods  can  at 
times  be  resorted  to  (§  65). 

A  system  of  n  total  differential  equations  in  n  -f- 1  variables  can  be 
written  as  a  system  of  ordinary  differential  equations  to  which  the 
methods  of  §  63  and  §  65  apply  (§  67). 

A  single  differential  equation  in  one  dependent  variable  of  higher 
order  than  the  first,  also  a  system  of  n  such  in  n  dependent  variables 
may  be  replaced  by  a  corresponding  system  of  differential  equations 
of  the  first  order,  to  which  at  times  the  special  methods  of  §  65 
apply  (§  68). 

It  is  almost  needless  to  add  that  if  each  of  a  system  of  equations 
involves  a  single  dependent  variable,  each  is  to  be  integrated 
independently  of  the  others.  Thus,  see  examples  i,  2,  3,  4  below. 

Ex.  1.  Find  the  path  traced  out  by  a  particle  moving  in  a  vacuum 
and  acted  upon  by  gravity  only,  if  it  is  given  an  initial  velocity  VQ  in 
a  direction  making  an  angle  a  with  the  horizontal  plane. 

Ex.  2.  If  the  particle  in  Ex.  i  moves  in  a  medium  which  exerts  a 
resisting  force  proportional  to  the  velocity,  find  its  path. 

Ex.  3.  A  particle  moves  about  a  center  of  attraction  varying 
directly  as  the  distance ;  determine  its  motion,  if  it  starts  to  move 


§69  SYSTEMS   OF   SIMULTANEOUS   EQUATIONS  163 

from  a  point  on  the  axis  of  x  at  a  distance  a  from  the  center,  and 
with  an  initial  velocity  VQ  making  an  angle  a  with  the  axis  of  x. 

[If  the  attracting  force  is  P,  and  r  is  the  distance  of  the  particle 
from  the  the  center  of  the  force,  the  equations  of  motion  are 

d*x  _      px 
d?~          r' 


d?  r 

In  this  case  P=#*r.'] 

Ex.  4.  If  the  force  is  a  repulsive  one,  study  the  motion  of  the 
particle  in  Ex.  3. 

Ex.  5.  A  solid  of  revolution  with  one  point  of  its  axis  of  symmetry 
fixed,  is  acted  upon  by  gravity  only.  Find  its  angular  velocity  and 
the  position  of  the  instantaneous  axis  of  rotation  in  the  body. 

[If  A,  B,  C  are  the  moments  of  inertia  of  the  body  with  respect 
to  the  principal  axes  of  the  momenta!  ellipsoid  about  the  fixed  point, 
and  /,  q,  r  are  the  components  of  the  angular  velocity  on  those  axes 
at  any  instant,  Euler's  equations  are, 


C  • —  =  o, 

since  B  =  A.~\ 

Ex.  6.  The  component  of  the  velocity  of  a  particle  parallel  to 
each  of  the  coordinate  axes  is  proportional  to  the  product  of  the 
other  two  coordinates.  Find  its  path,  and  the  time  of  describing  a 
given  portion  in  case  the  curve  passes  through  the  origin. 


CHAPTER   XI 

INTEGRATION  IN  SERIES 

70.  The  Existence  Theorem.  —  The  number  of  classes  of  differen 
tial  equations  that  can  be  integrated  by  quadratures  or  other  purely 
elementary  means  is  very  small,  compared  with  the  number  of  pos 
sible  classes  of  equations.  In  the  General  Theory  of  Differential 
Equations  it  is  proved  that  every  ordinary  differential  equation  with 
one  dependent  variable  (and  every  system  of  n  equations  with  n  de 
pendent  variables)  has  a  solution,  in  general,  involving  a  definite 
number  of  arbitrary  constants.  A  proper  understanding  of  the  proof 
of  this  theorem  implies  a  knowledge  of  the  Theory  of  Functions, 
which  is  not  assumed  here.  A  demonstration  of  the  theorem  will  be 
found  in  almost  any  book  dealing  with  the  subject,  presupposing  a 
knowledge  of  at  least  the  elements  of  the  Theory  of  Functions.* 

i°   For  an  equation  of  the  first  order  —  =  F  (^>7)>t   tne  theorem 

doc 
of  existence  of  an  integral  is  : 

*  Cauchy  (1789-1857)  was  the  first  to  prove  this  theorem.  In  fact  he  gave  two 
proofs  of  it,  which  have  become  classic.  For  a  demonstration  of  this  theorem  a  stu 
dent  familiar  with  the  elements  of  the  Theory  of  Functions  may  consult  among  other 
books,  Murray,  Differential  Equations,^.  190;  Schlesinger,  Differentialgleichungen, 
Chapter  I;  Picard,  Traite  d  Analyse,  Vol.  II,  Chapter  XI.  More  recently  Picard 
(1856-  )  gave  another  proof,  which  may  be  found  in  his  Traite  d' Analyse,  Vol.  II, 
p.  301,  and  Vol.  Ill,  p.  88,  and  also  in  the  Bulletin  of  the  New  York  Mathematical 
Society,  Vol.  I,  pp.  12-16. 

f  A  differential  equation  of  the  first  order  f\x,y,  -j-\  =  o  may  be  supposed  solved  for 

*   ««  4v.n*  . ...  1. .,  *u~  *....,   y 


2L%  so  that  it  takes  the  form^=  F(xty). 


164 


§70  INTEGRATION  IN   SERIES  165 

If  F(x,y)  is  finite,  continuous,  and  single-valued,  *  and  has  a  finite 
partial  derivative  with  respect  to  y  (see  Picard,  Vol.  II,  p.  292),  as 
long  as  x  and  y  are  restricted  to  certain  regions,  then  if  XQ  and  _y0  are 
a  pair  of  values  lying  in  these  regions,  we  can  find  one  integral  y,  and 
only  one,  which  will  take  the  value  yQ  when  x  takes  the  value  x^. 

In  the  proof  of  the  theorem,  jy  is  found  in  the  form  of  an  infinite  series 

yQ  +  e±(x  —  #0)  H-  c2(x  —  x0)2  H h  cn(x  —  x^f  -\ , 

which  series  satisfies  the  equation  when  substituted  in  it  for  y,  and 
besides  is  convergent  for  values  of  x  sufficiently  near  to  XQ.  By  the 
change  of  variable  x  —  x  —  xQt  the  differential  equation  takes  the 

dy —  x—     N 

form  g=^V*.>* 

and  the  solution  takes  the  form 

y  =7o  +  c\  *  +  ^2  x*  H h  cn  *n  4-  •••• 

Since  yQ  may  be  chosen  arbitrarily  (within  certain  limits,  however), 
we  see  that  in  the  case  of  a  differential  equation  of  the  first  order, 
one  arbitrary  constant  enters. 

Remark.  —  The  existence  theorem  gives  a  sufficient  condition  for  an  integral, 
and  moreover,  it  gives  a  form  in  which  the  integral  may  be  put.  But  this  condi 
tion  is  not  always  necessary.  Equations  for  which  the  conditions  of  the  theorem 
are  not  fulfilled  may  have  integrals.  In  general,  but  not  necessarily  always,  such 
integrals  will  then  not  be  developable  by  Taylor's  theorem,  or  they  will  not  be 
unique.  A  few  simple  examples  will  illustrate  this  : 

dy     y  y 

-j-  —  -,  where  -  becomes  indeterminate  for  x  =  o,  y  =  o,  has  the  solution  y  =  ex. 

*  Single-valued  is  used  in  the  broad  sense  here.  Although  F(x,y)  may  have 
several  values  for  a  single  pair  of  values  of  x  and>>,  it  will  be  said  to  be  single-valued 
when  x  andy  are  restricted  to  certain  regions  if,  having  selected  some  one  of  its  possible 
values  for  a  pair  of  values  of  x  and  y  in  their  respective  regions,  it  will  take  a  definite 
value  for  every  pair  of  values  of  x  and  y  in  their  regions. 

Thus,  while  F=  ±  V*  -\-y  has  two  values  for  every  pair  of  values  of  x  and  y,  if  we 
select  the  value  -f-  \/2  for  x  =  i,y  =  i,  ^will  have  a  definite  value  so  long  as  x  and/ 
are  restricted  to  regions  where  both  are  positive. 


\66  DIFFERENTIAL   EQUATIONS  §70 

Here  y  takes  the  value  o  for  x  =  o.  It  is  expressed  as  a  (finite)  Taylor  series, 
but  it  will  be  noted  that  c  is  undetermined  when  we  putjy  =  o  for  x  —  o  ;  that  is, 
unlike  the  cases  coming  under  the  existence  theorem,  there  is  an  indefinite  num 
ber  of  solutions  satisfying  the  initial  condition.  Moreover,  it  should  be  noted 
that  it  is  impossible  to  find  a  finite  value  for  c  that  will  enable  us  to  assign  a 
value  to  y  other  than  o  for  x  =  o. 

dy  _x  -f  y  x  -\-  y 

Again,  ,   —  >  where     ~     becomes  indeterminate  for  x  —  o,y  —  o,  has  the 

solution  y  —  x  log  x  +  ex.  Here  y  takes  the  value  o  for  x  —  o.  But  it  is  not  pos 
sible  to  express  the  integral  in  the  form  of  a  Taylor  series  in  powers  of  x.  In 
this  case  also  we  have  an  indefinite  number  of  integrals  for  the  one  initial  value 
o,  and  no  integrals  for  any  other  initial  value  of  y. 

2~dx  =  I  +  2X  has  for  solution  y  =  ^c  +  x  +  -*2-      +  2X  =o0  '  when  *  =  0> 

y  =  o.  In  order  that  y  =  o  for  x  =  o,  we  must  have  c  =  o.  We  have  then  the 
single  solution  y  =  Vx  +  x'2.  This  is  not  developable  by  Taylor's  theorem 
in  powers  of  x  however,  although  it  may  be  developed  in  powers  of  V-r. 

2°  If  we  have  a  system  of  n  equations  of  the  first  order  involving 
n  dependent  variables,  we  may  suppose  them  solved  for  the  deriva 
tives  of  each  of  these  variables  : 


,  *,—,*>), 


The  general  existence  theorem  says,  in  this  case,  that  if/i,/2,  •••/* 
are  all  regular,*  as  long  as  x,  y,  z,  •••,  w  remain  in  certain  regions, 
then  if  XQ,  y0t  £0,  •••,  WQ  are  in  these  regions,  a  single  set  of  functions 
y,  z,  •••,  w  can  be  found  to  satisfy  the  system  of  equations  and  to  take 
the  values  yQ)  ZQ,  •  •  •,  w^  respectively  when  x  takes  the  value  x^. 

*  For  definition  of  regular  see  the  second  footnote,  p.  203. 


§70  INTEGRATION   IN   SERIES  l6/ 

In  the  proof  of  the  theorem  these  are  found  in  the  form  of  the 
infinite  series 


y  =yQ  +  a^x  -  XQ)  +  az(x  -  -*0)2  H  -----  h  an(x  -  x0)n  +  •••, 
z  =  ZQ  +  &i(x  -  #„)  +  Az(*  -  -*o)2  H  -----  h  £«(*  -  -*o)n  +  —  , 

«/  =  w0  +  k^x  —  x0)  +  £2(*  —  -*o)2H  -----  h  £„(#  —  -*o)n  H  ----  , 

which  series  are  convergent  so  long  as  x  is  sufficiently  close  to  x0. 

As  before,  we  can  make  the  transformation  x  =  x  —  xQf  which  will 
then  give  our  series  as  power  series  in  x.  Here  y0,  z0,  •••,  WQ  may 
be  chosen  arbitrarily  (within  certain  limits).  Hence  we  see  that  a 
system  of  n  equations  of  the  first  order  with  n  dependent  variables, 
has  its  general  solution  involving  n  arbitrary  constants.  We  saw 
(§  68)  that  an  equation,  with  one  dependent  variable,  of  the  nth 
order  may  be  replaced  by  a  system  of  n  equations  of  the  first  order, 
hence  it  follows  at  once  that  the  general  solution  of  a  differential 
equation  of  the  nth  order  involves  n  arbitrary  constants.  In  the  way 
in  which  the  equivalent  system  of  equations  of  the  first  order  is 
found,  we  see  that  we  may  choose  for  these  arbitrary  constants  the 
values  which  the  dependent  variable  and  each  of  its  derivatives  up 
to  the  (»-i)st  take  for  a  given  value  of  the  independent  variable. 
Hence  these  values  are,  in  general,  at  our  disposal  in  any  given 
problem. 

Geometrically,  this  means  : 

Of  the  single  infinity  of  integral  curves  of  a  differential  equation  of 
the  first  order  and  degree,  a  single  curve  passes  through  a  given  point. 

Of  the  double  infinity  of  integral  curves  of  a  differential  equation 
of  the  second  order  and  first  degree,  a  single  curve  passes  through  a 
given  point,  in  a  given  direction. 

Of  the  triple  infinity  of  integral  curves  of  a  differential  equation  of 
the  third  order  and  first  degree,  a  single  curve  passes  through  a  given 
point,  in  a  given  direction,  and  having  a  given  curvature  at  that  point 


168  DIFFERENTIAL   EQUATIONS  §71 

71.    Singular  Solutions.  —  In  the  existence  theorem  of  the  previous 
section  stress  should  be  laid  upon  the  fact  that  the  existence  of  an 

dy 
integral  of  -j-  =  F(xty)  is  assured  only  as  long  as  F(x,y)  is  finite, 

(IOC 

continuous,  and  single-valued  in  the  region  of  (xQt  y0).  If,  now,  our 
equation  is  given  in  the  form 

dy 
/(*,  y,  /)  =  o,  where  y'  =  -£_, 

we  know  that,  in  general,  y1  is  expressible  as  a  finite,  continuous,  and 
single-valued  function  of  x  and  y  in  the  region  of  (xQ,yo),  and  takes 
a  perfectly  definite  finite  value  y'Q  for  x  =  x0,  y  =yQ.  It  can  be 
shown*  that  this  will  be  true  as  long  as 


, 


But  if  df 


then  the  expression  for  yf  in  terms  of  x  and  y  ceases  to  be  single- 
valued  in  the  region  of  (x0t  _>'0).  So  that  in  the  region  of  such  values 
for  x  and  y  the  existence  theorem  does  not  assert  the  existence  of  a 
solution.  As  a  matter  of  fact,  a  solution  does  not  exist  there  in 
general.  For  from 


we  can  solve  for  y  and  y\  thus 

^  =$(*), 

and  only  in  exceptional  cases  will 


*  A  proof  of  this  theorem  will  be  found  in  many  works  on  Analysis;  for  example  see 
Liebmann,  Lehrbuch  der  Differentialgleichungen,  p.  8. 


§72  INTEGRATION  IN  SERIES  169 

If  it  should  happen  that  <f>i(x)  =  ,  then  y  =  <f>(x)  is  a  solution 

ctoc 

of  the  equation ;  and  since  it  is  usually  distinct  from  the  general 
solution,  it  is  a  singular  solution.  Moreover,  it  is  identical  with  the 
singular  solution  we  encountered  in  Chapter  V. 

Similarly  in  the  case  of  differential  equations  of  higher  order  than 
the  first,  singular  solutions  may  occur.     Thus  if  there  is  a  solution  of 

/(x,y,  y',  •••,yn))  =  o  for  which  T-^-  also  vanishes,  this  solution  is,  in 
general,  a  singular  one.* 

More  generally,  a  system  of  equations  of  the  first  order 


dy         dw\  _ 
y'Z'"   W>~dx''"'~dx)~ 

(to  which  any  system  of  m  equations  in  m  dependent  variables  is 
always  reducible,  §  68)  may  have  singular  solutions  under  certain 
conditions.  See  Picard,  Vol.  Ill,  p.  52. 

72.   Integration,  in  Series,  of  an  Equation  of  the  First  Order. — 
If  the  equation 


cannot  be  integrated  by  any  of  the  known  elementary  methods,  the 
existence  theorem  tells  us  that  if  F(x,y,)  is  finite,  continuous,  and 
single-valued  in  the  regions  containing  x  =  o,  y  =  CQ  (there  is  no  loss 
in  assuming  x  =  o ;  since  this  amounts  to  presupposing  the  substitu 
tion  x  =  x  —  x0  to  have  been  made,  in  case  XQ  =£  o),  one  and  only  one 
solution  exists  which  takes  the  value  of  CQ  for  x  =  o.  But  this  solu- 

*  Liebmann,  loc.  cit.  p.  113 ;  Boole,  Differential  Equations,  p.  229. 


I/O  DIFFERENTIAL   EQUATIONS  §52 

tion  is  given  by  the  existence  theorem  in  the  form  of  an   infinite 
series.     In  actual  practice  we  assume 

(2)  y  =  %  <:»#" 


and  substitute  this  in  the  differential  equation  (i).  Then  equating 
coefficients,  we  may  calculate  as  many  of  the  <r's  in  (2)  as  we  please. 
The  existence  theorem  vouches  for  the  convergence  of  the  series  (2). 
Three  cases  may  arise  : 

i°  No  general  law  of  the  coefficients  in  (2)  shows  itself;  in  this 
case  we  can  only  approximate  the  solution  in  actual  practice.* 

2°  A  general  law  of  the  coefficients  in  (2)  appears  ;  then  we  can 
write  down  the  general  term  of  the  series,  which  is  equivalent  to  say 
ing  that  the  whole  series  is  known. 

3°  All  the  terms  after  a  certain  one  may  turn  out  to  be  zero  ;  in 
this  case  we  have  the  solution  in  finite  form. 

Remark.  —  Cases  2°  and  3°  seldom  occur  except  when  the  equation  can  be 
solved  directly.  So  that  this  method  of  integrating  equations  of  the  first  order  is 
not  of  great  practical  importance  for  the  mere  purpose  of  integrating.  But  for 
theoretical  purposes  it  is  of  the  greatest  importance.  It  may  be  noted  that  while 
every  linear  equation  of  the  first  order  can  be  solved  by  quadratures,  it  is  not 
always  possible  to  perform  these  in  terms  of  simple  functions.  In  such  cases  this 
method,  or  that  of  §  74,  will  apply  at  times. 

Ex.1.    ^  =  *+/. 
dx 

Here  x  +y2  is  finite,  continuous,  and  single-valued  for  all  values  of 
x  and  y. 

Put  y  =  CQ  +  c 


*  While  the  general  existence  theorem  tells  us  that  the  series  so  obtained  is  con 
vergent,  as  long  as  we  restrict  ourselves  to  proper  values  of  the  variables,  the  conven 
gence  may  be  slow  for  certain  values  so  that  the  degree  of  approximation,  even  after 
having  calculated  a  fairly  large  number  of  coefficients,  may  not  be  great.  This  will 
usually  be  true  for  values  of  the  variables  near  any  for  which  F(x,y)  ceases  to  be 
finite,  continuous,  or  single-valued. 


§72  INTEGRATION   IN   SERIES 

Substituting,  we  must  have 


c1+2c2x  +  3£aX?-\  -----  \-n 
Equating  coefficients,  we  have 


=  2 


3  4,  =  2  ^  4- ^i2  .'.  4j  =  -v0 


C 

4  <T4  =  2  <r</3  +  2  f^2  >'•  C\  —  ~C<?  + 

12 


(2  ^  +  1)^+1  =  2  ^2*  +  2  ^^-1  H 


Here  each  coefficient  can  be  calculated  in  terms  of  the  preceding 
ones,  and  consequently  in  terms  of  the  single  one  <r0.  We  can  cal 
culate  as  many  as  we  please,  but  no  general  law  shows  itself.  Our 
solution 


extended  as  far  as  we  care  to  calculate  the  coefficients,  is  only  an 
approximation  of  the  solution. 


y        2 

Ex-2-    - 


Here  — — — 5  i§  finite,  continuous,  and  single-valued  in  the  region 
of  (o,  <r0),  where  CQ  is  any  value  of  y. 


1/2  DIFFERENTIAL  EQUATIONS  §72 


Put   y  =  cQ  +  CiX  +  c.^  -\  -----  h  fnxn  +  •  •  •,   and    substitute    in   the 
equation,  after  having  cleared  of  fractions.     We  must  have 


*2-  4-  ^  *n  4- 
Equating  coefficients, 

^  =  2  <TO. 
2  <r2  =  2  <T!  =  4  <TO,  .-.  <r2  =  2  r0. 


The  general  law  seems  to  be  cn  =  2  CQ.     We  shall  prove  this  to  be 
the  case  by  showing  that  if  it  holds  for  cn  it  holds  for  cn+1.     For, 

(«  +  O^n-M  -  (»  -  1)^-1  =  2  ^. 
NOW  ^M_!  =  Cn  =  2  f0. 

/.   («  +  l)^n+l  =  2(«  -  l)r0  +  4  ^0  =  2(«  +  l)^(), 

or  <rn+1  =  2  f0. 

Hence  the  solution,  in  the  form  of  an  infinite  series,  is 

h  2  Xn        •••. 


[This  equation  can  be  integrated  by  the  methods  of  Chapter  II. 
Let  the  student  do  this,  and  compare  the  result  with  the  one  here 

obtained.] 

ay 

When  the  equation  —  =  F  (x,  y)  is  such  that  the  various  deriva- 
doc 

tives  of  F  can  be  readily  calculated  numerically  for  special  values  of 
x  and  y,  the  following  method  is  sometimes  found  practicable  : 

We  have  seen  that  the  solution  given  by  the  general  existence 
theorem  has  the  form 

X  —  XQ)  +  C<L(X  —  Xof  -\  -----  \-cn(x  —  x^-\  ----  . 


§73  INTEGRATION  IN   SERIES  1/3 

Using  the  general  form  of  the  Taylor  development  of  a  function, 
we  see  that  when 


jo  n    \xo 

we  obtain  that  solution  which  takes  the  value  yQ  for  x  =  .tr0. 

From  the  differential  equation,  we  have 


Differentiating  the  differential  equation  we  have 

#y=W+dFdy 
dx1       dx       dy  dx 

,  i    (d*y\         I   fdF\    .     I       fdF\ 

whence  e»~—  ,    -7^  }  =  —  -  (-T-)  +  —  ;'i(  T-  )• 

2  !  \4r/fl     2  !  \  flbr/0      2  !     \dyjo 

Differentiating  again,  we  find  c3  in  terms  of  ^  and  c^.     And  so  on. 
The  student  will  find  on  applying  this  method  to  the  examples 
above  that  it  works  very  readily  in  the  case  of  Ex.  i  . 

73.    Riccati's  Equation.  —  The  equation  studied  by  Count  Riccati 
(1676-1754),  and  to  which  his  name  has  been  given,  is  of  the  form 


where  b,  c,  m  are  constants.*     The  equation  in  Ex.  i,  §  72  is  of  this 
type.     For  certain  special  values  of  £,  c,  m  this  equation  can  be  inte- 

dy 
*  Frequently  the  equation  x  ~  —  ay-}-  j3y2  =  yxn  is  taken  as  the  type  of  a  Riccati 

equation.    This  is  obviously  reducible  to  the  other  by  the  transformation  z  =  xa,  y  =  uz. 


1/4  DIFFERENTIAL   EQUATIONS  §73 

grated  in  finite  terms.  (See  Ex.  4,  §  18  ;  also  Forsyth,  p.  170;  Boole, 
Chapter  VI  ;  Johnson,  Chapter  IX.)  But  in  general,  the  only  way 
to  get  the  solution  is  to  integrate  in  series. 

Riccati  equations  frequently  arise,  and  it  is  often  desirable  to  make 
use  of  the  properties  of  their  solutions  without  actually  knowing  the 
latter.  On  the  other  hand,  it  is  sometimes  possible  to  find  the  gen 
eral  solution  by  quadratures  or  by  merely  algebraic  processes,  when 
certain  information  is  at  hand.  The  following  properties  will  at 
times  prove  of  value  : 

We  shall  consider  a  more  general  form,  which  is  now  usually  con 
sidered  as  the  type  of  a  Riccati  equation, 


where  X0,  Xlt  X2  are  functions  of  x  or  constants. 

i°    If  a  particular  integral^  is  known,  the  substitution  y  =-  -f  yl 

z 

transforms  the  equation  to  —  -f  (Xi  +  2  y±  X2)z  =  —  X2,  which  is  linear, 

ClOC 

and  can   therefore  be  solved  by  two  quadratures    (§  13).      Hence 
we    have,  if  a  particular  integral  yl  is  known,  the  transformation 

y=--\-yl  gives  rise  to  a  linear  equation  in  z  which  can  be  solved  by 
z 

two  quadratures. 

2°  Since  the  form  of  the  solution  of  a  linear  equation  of  the  first 
order  is  z  =  y(x)  -f  CS(x),  that  is,  the  constant  of  integration  enters 
linearly,  we  have  that 


§73  INTEGRATION   IN   SERIES  i;$ 

Hence,  the  constant  of  integration  enters  bilinearly  in  the  general 
solution  of  the  Riccati  equation. 

3°   The  equation  y  =  a     ^     may  be  looked  upon  as  a  bilinear 


transformation  of  C  into  y,  which  latter  is  a  particular  solution  as  soon 
as  a  value  of  C  is  fixed.  Corresponding  to  any  four  values  of  C, 
say  Ci,  C2,  C3,  C4,  we  have  ji,  y»)  y&  y\-  Since  double  ratios  are  left 
unaltered  by  bilinear  transformations,  we  have 

\y\,  y*  y*>  y*\  =  \  c*  c*>  C*  C*\=*  constant. 
Hence,  if  y\^y^y^y^  are  any  four  particular  integrals,  the  function 
yi)  t-s  equai  to  a  constant  for  all  values  of  x. 


4°   As  a  direct  consequence  of  3°  it  follows  that  if  we  know  three 
particular  integrals  ylt  y2,  ys,  the  general  solution  is  given  at  once  by 

y-yi~   y*-y\* 


whence  y  =     3     2  ~       ~          2  ~        ,  i>e>  y  is  given  by  purely  alge- 

,    .  y2—yi  —  ^(y-2  —  ys) 

braic  means. 

_ 

5°    If  jVi  and  y2  are  two  known  particular  integrals,  put  z— 


y—y* 

and  take  the  derivative  of  the  logarithm  of  both  sides.     This  gives 


zdx      y—ydx      dx         y—y\dx 

Since  &  =  x0  +  Xly 

dx 


.  _  dy*\. 
dx) 


DIFFERENTIAL  EQUATIONS  §73 


^  have  ~ 

whence  z 


y-y* 

from  which  y  can  be  gotten  at  once.  Hence,  if  two  particular  inte 
grals  yl  and  y2  are  known,  the  transformation  z  =  *-—&  leads  to  an 

y-K 

equation  in  which  the  variables  are  separated,  so  that  it  can  be  solved 
by  a  single  quadrature. 

Properties  i°,  4°,  5°  call  attention  to  the  close  analogy  of  the  Ric- 
cati  equation  to  linear  differential  equations  ;  for  the  knowledge  of 
each  additional  particular  integral  brings  us  nearer  to  the  general 
solution.  Thus,  the  knowledge  of  a  single  particular  integral  enables 
us  to  reduce  the  problem  of  solving  a  Riccati  equation  to  one  of  solv 
ing  a  linear  equation  of  the  first  order,  that  is  to  one  involving  two 
quadratures;  the  knowledge  of  two  particular  integrals  enables  us  to 
find  the  general  solution  by  performing  a  single  quadrature;  while  a 
knowledge  of  three  particular  integrals  gives  us  the  general  solution  by 
a  very  simple  algebraic  process. 

6°   As  a  matter  of  fact,  the  substitution 


transforms  the  Riccati  equation  into  the  homogeneous  linear  equation 
of  the  second  order 


where  X\  =  ^* 
dx 


§74  INTEGRATION   IN   SERIES  177 

[Let  the  student  show  that,  conversely,  the  substitution  y  =  e^** 
transforms  a  homogeneous  linear  equation  of  the  second  order  into  a 
Riccati  equation.] 

74.    Integration,  in  Series,  of  Equations  of  Higher  Order  than  the 

First.  —  If  the  equation  when  solved  for  the  highest  ordered  deriva 

tive* 

<Py      „(         dv  < 
^=F(x,y?, 

dx*         \         dx 

is  such  that  the  successive  derivatives  of  F  can  be  readily  calculated 
numerically  for  special  values  of 

dy  d*y 


a  method  analogous  to  that  given  at  the  end  of  §  72  may  be  em 
ployed. 

The  solution  given  by  the  general  existence  theorem  being  in  the 
form 

y  =  <r0  +  *i(x  —  x0)  +  c£x  —  x^f  H  -----  h  cn(x  —  ^0)n  H  ----  , 

we  know  from  the  general  form  of  the  Taylor  development  of  a  func 
tion  that 

dy\  i  (<Py\  i  (d*y 


Now  the  general  solution  involves  three  arbitrary  constants,  and 
we  saw  (§  70,  at  the  end)  that  a  particular  solution  will  be  determined 

on  as  we  fix  the  values  of  y,  -/,  —  ^  for  x  =  XQ  ;   let  us  call  them 
.      „  dx  dx* 


as  soon 


*  To  fix  the  ideas  we  shall  illustrate  with  an  equation  of  the  third  order,  although 
this  method,  when  practicable,  applies  to  equations  of  any  order. 


DIFFERENTIAL   EQUATIONS  §74 


From  the  differential  equation  we  have  at  once 
d* 


Differentiating  the  equation  we  can  calculate  (  —  =?  )  in  terms  of 

Uk?Vo 


so  that  we  can  find  c±  in  terms  of  c^  tlt  cz,  cs.     And  so  on. 

When  this  method  does  not  work  readily,  we  may  employ  the  first 
method  of  §  72  as  there  given.  But  again,  only  in  comparatively  few 
cases  will  this  turn  out  to  be  practicable.  The  following  modification 
of  this  method  has  been  found  of  special  service  in  the  case  of  linear 
differential  equations  in  which  the  coefficients  are  polynomials  in  x, 
and  such  that  when  we  substitute  xm  for  y  in  the  left-hand  member, 
there  results  only  a  small  number  of  distinct  powers  of  x  (preferably, 
not  more  than  two).  In  the  case  of  a  Cauchy  equation  (§51)  there 
results  a  single  power  of  x,  and  the  equation  can  be  solved  by  purely 
algebraic  means.* 

If  we  take  the  equation 


the  result  of  putting  y  =  xm  in  the  left-hand  member  is 


*  Thus,  putting^  =  xm  in  the  left  hand  member  of  (i),  §  51,  we  get 

[kQm  (m  —  i)  •••  (tn  —  tr+  i)  +klm  (m—  i)—  (m  —  n  +  2)  H  -----  h  kn-\  m  +  £«]  x™. 

Equating  the  coefficient  of  xm  to  zero  and  solving  for  m,  we  get  in  general  n  distinct 
particular  solutions  and  therefore  the  complementary  function.  It  is  readily  seen  to  be 
the  same  as  that  found  in  §  51.  The  cases  of  equal  and  complex  values  of  m  can  be 
treated  entirely  analogously  to  those  for  linear  equations  with  constant  coefficients. 

In  some  respects,  the  Cauchy  equation  is  simpler  than  the  corresponding  linear 
equation  with  constant  coefficients.  But  for  actually  obtaining  its  solution,  especially 
when  there  is  a  right-hand  member,  it  is  usually  simpler  to  transform  it  to  a  linear 
equation  with  constant  coefficients,  as  was  done  in  $  51. 


§74  INTEGRATION   IN   SERIES  1/9 

Here  there  are  two  distinct  powers  of  x  which  differ  by  3,  and 
m  —  2  is  the  smaller  exponent.     Hence,  if  we  let 


y  =  c^ 

we  shall  have  a  solution  if 

i°  m  is  so  chosen  that  m(m  —  i)  =  o,  i.e.  m  =  o  or  i, 

2°  the  c's  are  so  chosen  that  the  rest  of  the  terms  cancel  each 
other  in  pairs,  i.e.  we  must  have 

(m  +  3)  (m  +  2)^  —  CQ     =o, 
(m  +  6)  (m  +  5)r2  —  *j     =o, 

O  +  3  r)  (m  +  3  r—  i)  ^  -  ^  =  o, 


For  m  —  o,  we  have  ^  =  — —  r0     =  —  <:0, 
3'2  3! 


7-1 

6-5  6  ! 


9-8'  g\ 

i       ,8=i'4-7- 

12  •  II  12  ! 


/•  -                   -  i  •4-7---[i  + 
c  —  — * — 

(sr) 


180  DIFFERENTIAL   EQUATIONS  §74 


is  an  integral.     Let  us  call  it  Ay1}  where  A,  like  CQ)  is  an  arbitrary 
constant. 

For  m  =  i  we  have  ^  =  -^—CQ   =  —  t0, 
4'3  4! 


TO  •  9  10  ! 


,_ 


is  also  an  integral.  Let  us  call  it  By^  where  B  is  an  arbitrary  con 
stant.  }>!  and  y2  are  obviously  linearly  independent.  Moreover, 
they  are  convergent  by  the  general  existence  theorem.  Hence, 
^=  Ayl  +  By z  is  the  general  solution,  since  it  contains  two  arbitrary 
constants. 

If  the  right-hand  member  of  the  equation  had  not  been  zero,  we 
would  have  proceeded  to  find  the  particular  integral  by  a  similar 
method.  Thus,  suppose  the  equation  had  been 


§74  INTEGRATION   IN   SERIES  l8l 

Since  the  result  of  putting  y  =  c%xm  in  the  left-hand  member  is 
ctfn  (m  —  i)xm~2  —  c^xm+l, 

ctfcm  -f-  ^xm+s  -\ -t-  crxm+3r  +  •  •  •  will  be  a  particular  integral,  pro 
vided  f0m(m  —  i)xm~2  =  2  x~s,  and  the  other  terms  that  arise  destroy 
each  other  in  pairs. 

The  first  of  these  will  be  true  if 

m  —  2   =  —  3,  .'.m=—i, 

and  f0m  (m  —  i)  =      2,  .•.  c0  =      i. 

The  other  terms  will  destroy  each  other  in  pairs  if 
fl==--^  .-.*•!  =  -*- 

2-1  2  ! 

^2  =  —  ^  .:t2  =  -2- 

5-4  5' 


Hence  a  particular  integral  is 


2!          5! 

The  above  example  suggests  a  general  method,*  in  case  the  result 
of  substituting  y  —  xm  in  the  left-hand  member  of  the  equation  gives 

*  As  mentioned  in  the  Remark,  §  72,  this  method  applies  to  linear  equations  of  the 
first  order  as  well  as  to  those  of  higher  orders. 


1  82  DIFFERENTIAL   EQUATIONS  §74 

rise  to  only  two*  distinct  powers  of  x,  say  f(m)  xh  -f  <£  (m)  x?+l, 
where  /  is  a  positive  integer. 

By  the  manner  in  which  f(ni)  and  <#>  (m)  arise,  it  is  obvious  that 
one  of  them  at  least  must  be  of  degree  n  in  m. 

I.   To  find  the  complementary  function  we  proceed  as  follows  : 

(a)  Suppose  /(w)  is  of  degree  n  in  m,  so  that/(#z)  =  o  has  n  roots, 
mlt  m2,  ms,  •••*#„,  all  distinct,  or  some  repeated. 

Letting  y  =  c^cm  +  c^xm+l  -\  ----  +  f^+rl  -\  ----  ,  we  have,  on  substi 
tuting  in  the  left-hand  member  of  the  differential  equation, 


\r  - 

+  ^  (m  +  rt)  x* 


This  will  be  zero  provided 

i°  /O)  =  o,  i.e.  m  =  ml}  m2,  ->,  mn  ; 

_l^+[r_I]/)=0,  for 


4-  [r- 


*  An  analogous  method  can  be  deduced  in  the  case  of  a  larger  number,  but  this  will 
not  be  done  here. 


§74  INTEGRATION   IN   SERIES  183 

To  each  value  of  m  corresponds,  then,  in  general,  a  particular 
integral.*  If  any  c  vanishes,  all  that  follow  do  so,  and  that  integral 
appears  in  finite  form. 

If  two  values  of  m  are  equal,  of  course,  the  same  particular  integral 
will  correspond  to  these.  Moreover,  if  two  of  the  m's  differ  by  an 
integral  multiple  of /,  say  m2=ml-\-g/)  then  corresponding  to  the 
smaller  value  mlt  the  coefficient  cg  will  be  infinite  [since  /(m2) 
=/(mi+gt)  =o]»  unless  the  numerator  is  also  zero.  Hence  the 
method  here  given  gives  us  only  as  many  particular  solutions  in 
general,  as  there  are  distinct  roots  of /(m)  =  o,  whose  differences 
are  not  multiples  of  /.  The  remaining  solutions  must  then  be  sought 
by  a  modification  of  our  process.  | 

If  f(m)  is  of  lower  degree  than  n,  while  the  above  method  may 
lead  to  infinite  series  which  will  satisfy  the  equation,  the  general 
theory  gives  us  no  assurance  that  they  are  convergent.  So  that, 
unless  the  solutions  come  out  in  finite  form,  it  is  best  to  make  use 
of  the  fact  that  if  f(ni)  is  not  of  the  degree  n,  <J>(m)  is. 

(&)  If  <t>(m)  is  of  degree  n  in  m,  <f>(m)  =  o  will  be  satisfied  by  n 
values  of  m,  say  m\,  m'2,  •••  m'n. 

Letting  y  =  c&m  +  c_.^~l  +  <r_2*w-21  H h  c^xm-*  -\ ,  we  have 

on  substituting  in  the  left-hand  member  of  the  equation, 


*  It  should  be  noted  that  corresponding  to  any  m  which  is  not  a  positive  integer,  we 
have  a  solution  which  is  not  a  power  series,  but  x™  multiplied  by  a  power  series. 

Although  the  general  existence  theorem  no  longer  applies  here,  because  the  coeffi 
cients  in  the  equation,  when  the  leading  coefficient  is  made  unity,  cease  to  be  finite 
for  x  =  o,  the  general  theory  of  linear  equations  assures  us  of  the  convergence  of 
the  series  for  certain  values  of  x.  (Schlesinger,  Di/erentialgleichungen,  \  24.) 

t  In  this  case  the  particular  solutions,  which  the  general  method  fails  to  give, 
usually  involve  logarithmic  terms.  Thus  see  Ex.  2. 


1  84  DIFFERENTIAL   EQUATIONS  §  74 

+  c_r+l  <f>(m  -  [r  -  i]  /X^1-2*  +  c_r+J(m  -  [r  - 


This  will  be  zero  if 


=  o,  *>.  w  =  m\,  m'2} 


m 


1,2,3,  ...,«,    i 

=  m\,  m'2)  .•-,  w'.J 


—  \r-\~\l)  •••  0(w  —  2  /)<£(>  -/) 


To  each  value  of  #2  corresponds,  in  general,  a  particular  integral 
of  the  form  xm(cQ  +  ^  x~l  +  r_2  ^~2J  +  ••  •  +  ^r  -^~"  +•••)• 

Here  xm  is  multiplied  by  a  power  series  going  according  to  nega 
tive  ascending  powers  of  x.  The  latter  reduces  at  once  to  an 

ordinary  power  series  in  /  if  we  put  /=  -•      Of  course,  if  any  of  the 

roots  of  <f>(m)  =  o  are  repeated,  or  differ  by  a  multiple  of  /,  the  num 
ber  of  integrals  obtained  by  this  method  will  be  less  than  n. 

It  should  be  noted  that  the  integrals  found  by  methods  (a)  and 
(b)  in  case  both  f(m)  and  <£(w)  are  of  the  flth  degree,  are  not 
distinct.  In  general  (but  not  necessarily),  they  are  different  in 
form,  but  only  n  of  the  functions  defined  by  them  can  be  linearly 
independent.* 

*An  infinite  series  defines  a  function  for  those  values  of  the  variable  only  for 
which  the  series  is  convergent.  The  series  found  by  methods  (a)  and  (b),  being 
developments  in  the  region  of  the  origin  and  of  oo  respectively,  may  not,  and  fre 
quently  do  not,  converge  for  the  same  values  of  x.  Hence  if  the  series  are  infinite,  it 
is  usually  impossible  to  compare  them.  But  if  the  functions  represented  by  each  set 
can  be  "  continued"  into  the  region  of  the  other,  then  a  linear  relation  will  be  found 
to  exist  among  any  n  +  i  of  them. 


§74  INTEGRATION   IN   SERIES  1^5 

II.  To  find  the  particular  integral  in  case  the  right  hand  member 
is  a  power  of  x,  say  Ax",  we  proceed  as  follows  : 

If  f(m)  is  of  degree  ;/,  we  must  have,  using  the  results  found  in 
connection  with  method  (a)  above  : 

i°  h  —  s  .  Since  h  is  a  linear  function  of  m,  this  will  determine  a 
single  value  of  m,  say  mg. 

2°   c0/(m.)=A. 

3°  The  remaining  coefficients  are  determined  as  in  (a)  except 
that  now  ms  is  used  for  m. 

This  method  will  be  in  default  when  ./(*«,)  =  o. 

Or  using  the  results  found  in  connection  with  method  (<£)  *,  we  get 
the  solution  by  putting  : 

i°    h  +  /=  s.     This  will  determine  a  value  ms'  of  m. 

2°  r0<KO  =  A. 

3°  The  remaining  coefficients  are  determined  as  in  (b)  except 
that  now  ms'  is  used  for  m. 

If  <j>(nig)  =  o  at  the  same  time  that  f(ms)=  o,  the  particular  inte 
gral  will  not  be  of  the  form  here  sought.  Other  means  will  have  to 
be  used  to  find  it. 

For  a  perfectly  general  method  for  finding  the  particular  integral 
see  Schlesinger,  §54. 


Substituting  xm  for  y  in  the  left-hand  member,  we  have 

m(m  —  4)^TO~1  —  \jn(ni—  i)  —  2\xm. 

*  Whether  4>(m)  is  of  degree  n  or  not.  The  objection  to  using  method  (a)  or  (6) 
in  case  f  (m)  or  <b(m)  is  not  of  degree  n  is,  that  if  the  solution  comes  out  in  the  form 
of  an  infinite  series,  the  latter  need  not  be  convergent.  If  the  solution  appears  in 
finite  form,  the  method  applies  perfectly  well;  if  not,  the  convergence  of  the  series 
must  be  looked  into.  In  the  case  of  the  example  worked  out  above,  if  the  right-hand 
member  had  been  x,  method  (a)  would  have  given  the  particular  integral  in  the  form 
of  an  infinite  series,  which  turns  out  to  be  yl  —  i.  On  the  other  hand,  method  (b^ 
gives  —  i  for  the  particular  integral  at  once. 


1 86  DIFFERENTIAL  EQUATIONS  §74 

.-./=!, 

<j>(m)  =  —  (m  4-  i)(*»  —  2). 

Since /(w)  is  of  degree  2,  we  shall  use  method  (a). 
Then  w  =  o  or  4, 


and  cr *,  -  *(»»+^-'1)^r_1  =  ^  +  ^~3, 

For  m  =  o  we  have  i  —  -?         2 

*i  = -4)  =-^o, 

1  —  4          3 


— *i  —  •— <-o> 
2-4         3 


3  -4 

=o, 


/.  r0[  i  4-  -j:  4-  -^2 ),  or  ^(3  4-  2jc  4-  ^2)  is  an  integral.     Let  us  call 
V        3        3    / 


For  w  =  4  we  have          _  i  4-  i 

^"l  —  ^"Q  —  2  ^"oj 


—  — 


•  • 


§74  INTEGRATION  IN   SERIES  l8/ 

.'.  <r0(*4  -h  2  .X5  +  3  *6  H  -----  M*n+3H  ----  )  is   an   integral.       Let  us 
call  it  Byz. 

Therefore  Ay±  -f  Byz  is  the  complementary  function. 

Since  <f>(m)  is  of  the  second  degree,  we  can  also  use  method  (&). 

Then  m  =  2  or  —  i, 


and 
anu 


•  /  /  \  — r-fi 

<p(m  —  r)  m  —  r  —  2 

For  m  =  2  we  have 


2 


-  1  =  3  ^o, 


+  5-^~2H  -----  h  nx?~n+  -..)  is  an  integral. 
Let  us  call  it  A'yJ. 
For  m  =  —  i  we  have 


= 
4 


-2-3  4 

=  ~3~4^_2  =  - 
-3-3  4 


/.  ^(4  x-1  +  5  ^c-2  +  6  *-3  +  .-  +  («  +3)  ^-n  +  — )  is  an  integral. 

4 
Let  us  call  it  £'y2'. 

Hence  Ay±  +B'y£  is  the  complementary  function. 
Here  it  turns  out  that  yj  —  y2'  =y^ 


1  88  DIFFERENTIAL   EQUATIONS  §74 

To  find  a  particular  integral,  we  consider  each  term  of  the  right- 
hand  member  separately.  * 

For  the  term  x  we  have,  putting 

cQm(m  —  4)  xm~l  =  x, 

m  =  2  and  c0  =  —  —  . 
4 

r~  I 


For  m  —  2  cr  = 

r—  2 

.•.  d  =  o,  and  all  that  follow  are  zero. 

Hence  —  -  x2  is  a  particular  integral  corresponding  to  x. 

4 
Corresponding  to  3  x4,  we  have,  putting 


m  =  5  and  <TO  =  -• 


For 


Hence  -  [2  x5  +  3  x6  +  4  x1  +  •••  +  («  +  i) *"+4+ 


§  74  INTEGRATION   IN    SERIES  1  89 

is  a  particular  integral  corresponding  to  3  je4.     Comparing  this  with 
y2  above,  we  see  that  it  is  equal  to  —  (y2  —  -*4)- 


Hence  a  particular  integral  is  —  -^- 
The  complete  solution  is  then 


The  result  of  putting  y  =  xm  in  the  left-hand  member  is 
mz  xm~l  -f-  xm. 

.•./=!, 

/(m)  =  m2,  4>(m)  =  i. 
Hence  method  (a)  applies.      m  =  o,  o, 


since  m  =  o  is  the  only  choice  for  m. 


is  an  integral.     Let  us  call  it  cy^ 

*This  form  could  have  been  obtained  at  once  by  equating  c^(m)xm  to  $x*. 


1  90  DIFFERENTIAL   EQUATIONS  §74 

To  find  a  second  integral,  Iety=y1v  +  w,  where  v  and  w  are  two 
functions  still  to  be  determined.  Since  the  single  requirement  of 
having  this  value  of  y  satisfy  the  equation  is  imposed  upon  these  two 
functions,  a  second  relation  between  them  may  be  assumed.  We 
shall  do  this  so  as  to  simplify  our  work  as  much  as  possible. 

Substituting  this  value  of  y  in  the  equation,  and  remembering  that 
yi  is  an  integral,  we  get 


dw 
— 
dx 


d?v      dv\  ,         dv,  dv 
_  +  _    +2<*-^  —  = 
dx*      dx)  dx  dx 


Assume  now  that  x—9  +  —  =  o 

dx*      dx 


(this  is  the  second  relation  at  our  disposal).     Then 
v  =  A+J3  log  x. 

And  our  equation  to  determine  w  is 


+         _a2t 

dx*      dx  dx 


2!        2!3!      3!4!  »!(*+x)l 

Letting    w  =  t0  +  c^x  +  ^  H h  ^^n  +  —,' 

whence        ™  =  cl-\-  2 

d2w 


$  -4  INTEGRATION   IN   SERIES  1 9! 

we  have  on  equating  coefficients 


=(- I)' 


Since  we  are  looking  for  as  simple  a  particular  integral  as  possible, 
there  will  be  no  loss  in  assuming  <r0  =  o ;  then 


£9  =  — 


2    3 


Hence  the  general  solution  is 


-+- 

23 


DIFFERENTIAL   EQUATIONS  §75 


Ex.4.    2*2- 

ax*        dx 


75.   Gauss's  Equation.     Hypergeometric  Series.  —  The  integration 
of  the  equation 


az 

where  A,  B,  C,  D,  E,  F  are  constants  and  IP  —  ^AC^o,  leads  to 
a  remarkable  series  exhaustively  studied  by  Karl  Friedrich  Gauss 
(1777-1855).  This  series  and  its  differential  equation  were  dis 
covered  by  Euler  (1707-1783).  Putting  z  =  ax  +  b,  we  have 


d)x  +  Air  +  Bb+C}+  (D'x  +  E")    - 
H'OC  doc 


where  D',  E'  ',  Fl  are  constants. 
Choosing  a  and  b  so  that 


and  2  Ab  +  B  =  —  Aa  =£  o, 

and  dividing  by  the  coefficient  of  (x2  —  x)    J,  we  have 


where  P,  Q,  R  are  constants. 


§  75  INTEGRATION  IN   SERIES  1  93 

Substituting  xm  for  y  in  the  left-hand  member,  we  have 

-  m(m  -  i  -  Q)xm~l  +  [>2  -  (i  -  P)m  +  R~\xm. 

Putting  Q  =  -y,  i-P=-(a 

our  equation  takes  the  form 


and  the  result  of  substituting  y  =  xm  in  the  left-hand  member  is 


=—  m(m  —  i  4-y)>  </>(^)  =  ( 

Using  the  method  (d!)  of  §  74,  we  have 
m  =  o  or  i  —  y. 


_ 
/(m  +  r)  (OT  +  r)(«  +  r  +  y_I) 


For  w  =  o,  we  have 

_<*./? 
=  —  - 

I   -y 


2-y(y+l) 


This  is  usually  referred  to  as  Gauss's  equation. 


194  DIFFERENTIAL   EQUATIONS  §75 

Putting  t0=  i}  we  have  the  particular  integral 
,1=1+«^^« 

I-y  I-2 


j  -.  -^  | 

I  -2  -3  •••  n.y(y+i).>.(y  +  n-l) 

This  is  the  hypergeometric  series,  and  is  usually  represented  by 
F(a,  ft  y,  #). 

For  m  =  i  —y,  the  student  should  show  that  the  integral  is  cQy^ 
where  jy2  =  x^F(a.  —  y  +  i,  /3  —  y  +  i,  2  —  y,  #). 

If  a  or  ft  is  a  negative  integer,  while  y  is  not,  j^  reduces  to  a 
polynomial. 

If  y  is  a  negative  integer  (including  zero),  say  y=  —  g^o,  while 
neither  a  nor  /?  is  one  of  the  integers  from—  g  to  o,  the  coefficients 
in  yl  beginning  with  the  gth  become  infinite.  So  that  this  form  for 
_ft  cannot  be  used.  (Another  form  involving  log  x  can  be  found, 
which  together  with  y2  will  give  the  general  solution.) 

If  y  =  —  g^o,  where  g  is  an  integer,  and  a  or  (3  is  one  of  the 
integers  from  —  g  to  o,  _ft  reduces  to  a  polynomial  (excepting  in  the 
case  when  a  or  (3  =  y  \  in  this  case  a  factor  in  the  numerator  and 
one  in  the  denominator  of  every  coefficient  beginning  with  the  gih 
term  vanish  ;  here  these  zero  factors  neutralize  each  other,  and  the 
result  obtained  by  striking  out  these  zero  factors  gives  us  an  available 
form  for  .ft.*)  In  this  case,  although  the  two  roots  of  /(/«)  —  o 
differ  by  an  integer,  no  logarithmic  term  enters  in  the  general 
solution. 

If  y  is  a  positive  number,  say  y  =g  >  o,  and  neither  a  nor  ft  is 
one  of  the  integers  from  i  to  g,  the  coefficients  in  y2  become  infinite. 
In  this  case  a  new  form  for  y2  involving  log  x  can  be  found. 

*  See  Schlesinger,  Differ  entialgleichungen,  \  34. 


§75  INTEGRATION   IN   SERIES  195 

If,  however,  y  =g  >  o  where  g  is  an  integer,  and  a  or  fi  is  one  of 
the  integers  from  i  to  g,  y.2  reduces  to  a  polynomial  (with  the  excep 
tional  case  of  a  or  /8  =  y,  which  is  handled  in  the  same  manner  as  the 
exceptional  case  for  y  a  negative  integer  or  zero).  So  that  in  this  case 
also  no  logarithmic  term  enters  in  the  general  solution. 

Using  the  method  (fr)  of  §  74,  let  the  student  show  that 


are  a  pair  of  linearly  independent  integrals. 

If  y1  and  y.2  are  infinite  series  with  finite  coefficients,  they  will  be 
convergent  for  all  values  of  x  less  than  i  in  absolute  value,  and  diver 
gent  for  all  values  of  .#  greater  than  i  in  absolute  value;  while  yj 
and  yj  are  convergent  as  long  as  x  is  greater  than  i  in  absolute  value, 
and  divergent  as  long  as  x  is  less  than  i  in  absolute  value. 

The  hypergeometric  series  may  at  times  represent  well-known 
functions.  Thus  let  the  student  show  that 

Ex.  1.    F(—n,P,p,  —  x)  =  (i+  x)n  for  /?  any  constant. 
Ex.  2.    xF(i,  i,  2,  —x)  =  log  (i  +  x). 


Ex.3. 

Ex.  4.    Limit  a  B=OD  xF  (a,  Q,  &  ,  --  \\  =  sin  x. 

\         2 


Ex.  5.   Express  as  hypergeometric  series  the  following  functions  : 


I  *~~  OC 

For  further  examples  see  Gauss,  Collected  Works,  Vol.  Ill,  p.  127 


CHAPTER   XII 
PARTIAL  DIFFERENTIAL  EQUATIONS 

76.  Primitives  involving  Arbitrary  Constants.  —  Partial  differen 
tial  equations  may  be  obtained  from  primitives  involving  either  arbi 
trary  constants  or  arbitrary  functions.  Thus,  consider  the  family  of 
spheres  of  fixed  radius  R,  with  their  centers  lying  in  the  plane  of 
2=0.  The  equation  of  this  family  is  obviously 

(i)  (x-ay  +  (y-t)2+z2  =  R2, 

where  a  and  b  are  arbitrary  constants. 

We  shall  consider  x  and  y  as  independent  variables,  and  shall  put 

dz          dz  d2z  d2 


^~  —     y  ^T  —     >  ^    o  —     >  ^     ^~~  —     >  T~9 

dx          By          dx2          dxdy          dy2 

Differentiating  (i)  with  respect  to  x  andjy  respectively,  we  get 

(2)  x  —  a  +  zp  =  o, 

(3)  y- 


Eliminating  a  and  b  from  (i),  (2),  (3)  we  get 
(4)  **(/  +  ?*+  i)  =  *«, 


which  is  known  as  the  differential  equation  corresponding  to  the  primi* 
tive  (i)  ;  on  the  other  hand  (i)  is  said  to  be  a  solution  of  (4). 

136 


§76  PARTIAL  DIFFERENTIAL  EQUATIONS  197 

Perfectly  generally,  if  we  start  with   any  relation  involving  two 
arbitrary  constants  and  three  variables,  of  which  two  are  independent, 

(i)  <j>(x,y,z,a,fi)  =  o, 

where  z  is  taken  as  the  dependent  variable,  we  get  on  differentiating 
with  respect  to  x,  and  then  to  y 


We  now  have  three  equations  from  which  to  eliminate  a  and  £, 
Doing  this,  there  results 

(4)  f(x>y>z,P,q)  =°> 

a  differential  equation  of  the  first  order  which  has  (i)  for  its  primitive. 
If  the  primitive  involves  more  than  two  arbitrary  constants,  mor? 
than  three  equations  will  be  necessary  to  eliminate  these,  so  that  if 
only  two  independent  variables  are  involved,  the  resulting  differential 
equation  will  be  of  higher  order  than  the  first.  Thus  consider 

(5)  ^  +  ^/  +  ^2=i. 

Differentiating  with  respect  to  x  and  y  respectively,  we  have 

(6)  ax  +  czp  =  o, 
(7) 


We  have  now  only  three  equations  from  which  to  eliminate  a,  b,  c. 
Hence  we  must  differentiate  again.     Differentiating  (6)  with  respect 
to  x  we  have 
(8) 


198  DIFFERENTIAL   EQUATIONS 

Eliminating  a,  byc  from  (5),  (6),  (7),  (8),  we  have 


§76 


X  O  Zp 

0  y        zq 

1  o         zr 


o,  or 


(9)  (xzr  +  x/  -  zp)  =  o, 

which  is  a  differential  equation  having  (5)  for  its  primitive.     Had  we 
differentiated  (6)  with  respect  to  y,  we  would  have  gotten 

(8')  czs  +  cpq  =  Q-9 

whence,  on  eliminating  a,  b,  c  from  (5),  (6),  (7),  (S1), 

(9')  zs+pq  =  o, 

which  is  also  a  differential  equation  having  (5)  for  its  primitive.    Again, 
differentiating  (7)  with  respect  to  y,  we  get 

(8")  t 


Eliminating  a,  b,c  from  (5),  (6),  (7),  (8"),  we  have 
(9") 


which  is  also  a  differential  equation  having  (5)  for  its  primitive. 

Since  (9),  (9'),  (9")  are  all  of  the  second  order,  there  is  no  choice 
among  them,  and  all  may  be  said  to  be  differential  equations  belong 
ing  to  (5)*. 

*  Attention  should  be  called  to  the  fact  that  there  is  no  such  ambiguity  in  the  case 
of  the  equation  of  the  first  order  belonging  to  a  primitive  involving  two  constants. 
Since  there  are  k  -\-  i  ki\\  derivatives  of  z  depending  on  the  number  of  times  we  differen 
tiate  with  respect  to  x  and  to^,  it  is  clear  that,  in  order  that  a  primitive  should  give 
rise  to  a  unique  partial  differential  equation  of  the  «th  order,  the  number  of  arbitrary 
constants  it  contains  must  be  equal  to  the  sum  of  all  the  integers  from  2  to  n  +  I,  i.e. 
n(n  -j~3) 

2 


§77 


PARTIAL  DIFFERENTIAL   EQUATIONS 


199 


Form  the  differential  equations  having  the  following  equations  for 
primitives,  a,  b,  c  being  the  arbitrary  constants  to  be  eliminated  : 

Ex.1.  (x-ay  +  (y-a)2  +  z2  =  P. 

Ex.  2.  <*( 

Ex.  3.  z 

Ex.  4.  z 

Ex.5.  (x-ay+(y-t> 

Ex.  6.  z  =  ax  +  by  +  £jey. 

Ex.  7.  0#  -f-  £y  +  fls;  =  i  . 


77.    Primitives  involving  Arbitrary  Functions.  —  Let  u  and  0  be 
two  known  functions  of  x,  y,  z,  and  suppose  we  have  the  arbitrary 
relation 
(i)  4>(u,  v)  =  o. 

Differentiating  with  respect  to  x  and  y  respectively,  we  have 


In  order  that  (2)  and  (3)  may  hold  simultaneously,  we  must  have 


du      du 

"7:  I  ^  J 


dv      dv 
Bx     d* 


du      du 


dv  .  dv 
By     tot* 


=  o,  or 


(4) 


200 
where 


DIFFERENTIAL   EQUATIONS 


§77 


du  du 

dz  By 

dv  dv 

dz  dy 


du  du 

dx  dz 

Bv_  dv_ 

dx  dz 


du  du 

dy  dx 

dv_  dv_ 

dy  '  dx 


This  is  a  linear  partial  differential  equation  of  the  first  order.     By 
a  linear  partial  differential  equation  of  the  first  order  we  mean  one 
that  is  linear  in  the  derivatives  of  the  dependent  variable  (the  way  in 
which  the  dependent  variable  itself  enters  playing  no  role).* 
If  u  and  v  are  two  known  functions  of  x,  y,  z,  and  we  have 


(i)  f\_x>  y>  z>  <t>(u) 

where/  is  some  known  function,  but 
then,  on  differentiating,  we  have 


=  o 


and  \j/  are  arbitrary  functions, 


Since  (2)  and  (3)  introduce  two  new  functions  <j>'(u),  *l/'(v),  five 
equations  are  necessary  to  eliminate  the  four  functions  <t>(u),  t/f(z>), 

*'(«),  *'(»)• 

We  must  differentiate  (2)  and  (3),  which  will  give  rise  to  three 

new  equations  involving  second  derivatives  of  z,  and  the  new  func 
tions  <f>"(u)  and  $"(v).  In  all  we  have  now  six  equations  from  which, 
in  general,  it  is  not  possible  to  eliminate  the  six  arbitrary  functions. 
If  such  is  the  case,  we  must  differentiate  again,  this  time  obtaining 
four  new  equations,  involving  third  derivatives  of  z,  and  the  two  new 


*  The  student  will  note  the  difference  between  this  definition  and  that  given  for  a 
linear  ordinary  differential  equation  of  the  first  order  (§  13)  .    See  also  $  87. 


§77  PARTIAL  DIFFERENTIAL   EQUATIONS  2OI 

functions  <j>'"(u)  and  \j/'"(v).  That  is,  we  have  now  ten  equations, 
from  which  the  eight  functions  can  be  eliminated,  in  general,  in  two 
distinct  ways.  Hence  we  are  led  in  the  general  case  to  two  differen 
tial  equations  of  the  third  order. 

In  the  case  of  special  forms  of  the  function/,  we  can  eliminate  the 
six  functions  <f>(u),  <£'(V),  <£"(#),  $(v}>  $'(v\  V(v)  fr°m  tne  first  six 
equations  arising  in  the  above  process.  Thus,  for  example,  suppose 
/=  w  —  [<£(#)  -f  $(v)~]  =  o,  where  w  is  a  known  function  of  x,  y,  z. 

Then 


These  last  two  equations  involve  only  <£'(V)  and  \l/\v),  and  not 
<£(//)  and  \l/(v).  Hence,  since  the  three  equations  gotten  by  differen 
tiating  these  involve  only  <£'(»,  <£"(*),  $'(v),  t"(v),  these  four  func 
tions  can  be  eliminated  from  the  five  equations  in  which  they  enter 
and  a  single  differential  equation  of  the  second  order  arises  which 
has  w  =  <£(#)  +  \l/(v)  for  its  primitive  for  any  form  of  the  functions 
<f>  and  i/r.* 

Find  the  differential  equations  arising  from  the  following  primi 
tives  : 


Ex.1.   4>( 

Letting  x  +y  +  z  =  u,  x2  +  f  +  z?  =  vt  we  have  <j>(ut  v)  =  o. 

*  The  five  equations  are  linear  in  </>'(«),  </>"(»),  $'(v),  V(v).  Hence  the  elimina 
tion  can  be  effected  readily.  Moreover  r,  s,  t  enter  linearly  also,  and  in  such  a  way 
that  the  resulting  differential  equation  is  also  linear  in  them,  that  is,  it  has  the  form 
Rr  +  Ss  +  Tt=V,  where  Rt  St  T,  V,  are  functions  of  x,  y,  z,  p,  q. 


2O2  DIFFERENTIAL  EQUATIONS  §78 

Differentiating,  we  get 


OF 

Ex.  2.    </>  f  z~  —  xy,  2-  )  =  o. 

\  *J 

Ex.  3.    <£(*2  -f  /,  z  -  xy)  =  o. 
Ex.4.   z  =  <l>(x+y)+\p(x-y). 
Ex.5.   z=» 


78.  Solution  of  a  Partial  Differential  Equation.  —  Since  a  primitive 
which  gives  rise  to  a  differential  equation  is  obviously  a  solution  of 
that  equation,  we  see  that  arbitrary  functions  as  well  as  arbitrary  con 
stants  may  enter  into  the  solutions  of  partial  differential  equations.  By 
the  general  existence  theorem*  for  partial  differential  equations,  it  is 
seen  that  every  partial  differential  equation,  or  system  of  such  equa 
tions,  has  a  solution  containing  a  definite  number  of  arbitrary  func 
tions.  As  an  arbitrary  function  may  contain  an  indefinite  number  of 
arbitrary  constants,  a  solution  involving  an  arbitrary  function  is  much 

*  This  theorem  is  due  to  Cauchy,  as  is  that  for  ordinary  differential  equations. 
Proofs  of  the  theorem  have  also  been  given  by  Darboux  and  Mme.  Sophie  de  Kowa- 
lewski,  among  others.  The  proof  of  the  latter  is  the  most  readily  followed  and  is  the 
one  usually  given.  See  Goursat-Bourlet,  Equations  aux  Derivees  Partielles  du  Premier 
Ordre,  Chapter  I  ;  also  Picard,  Traite  d'  Analyse,  Vol.  II,  p.  318. 


§78  PARTIAL  DIFFERENTIAL  EQUATIONS  2O3 

more  general  than  one  that  contains  any  fixed  number  of  arbitrary 
constants.  We  shall  speak  of  the  solution  given  by  the  existence  theo 
rem  (which  is  a  solution  involving  one  or  more  arbitrary  functions) 
as  the  general  solution. 

The  mere  statement  of  the  existence  theorem  for  a  system  of  partial  differential 
equations  of  any  order  is  quite  complicated.  We  shall  give  here  simply  a  state 
ment  in  the  cases  of  a  single  equation  of  the  first  and  second  orders  in  three 
variables : 

i°  Consider  the  equation  f(x,  y,  z,  p,  q)  —  o.  We  shall  suppose  that  p 
actually  appears.*  Solve  for  it,  so  that  the  equation  may  be  supposed  to  have 
the  form 

/=  F(x,y,ztq). 

The  existence  theorem  tells  us  that  if  F  (x,  y,  z,  a}  is  regular  t  in  the  regions 
ofx  =  XQ,  y  =  yQ,z  =  ZQ,  q  =  q§,  and  if  <t>(y)  is  any  arbitrarily  chosen  func 
tion  of  y,  regular  in  the  region  of  y§,  and  such  that  0(jo)  =  zo>  0'Cyo)  =  £o» 
there  exists  one  and  only  one  solution  z  =  \f/(x,y),  which  is  regular  in  the  regions 
of  x  =  XQ,}>  =  }>Q,  and  which  reduces  to  z  =  <f>(y}  for  x  =  XQ. 

Geometrically  this  means  that  given  any  curve  z  =  0O')  *n  the  plane  x  =  XQ 
there  exists  one  and  only  one  surface  (in  any  region  for  which  there  are  no  sin 
gular  points  |  of  the  differential  equation}  passing  through  that  curve. 

This  can  be  generalized.  By  a  proper  choice  of  coordinates  it  can  be  shown 
that  one  integral  surface,  and  only  one,  can  be  found  passing  through  any  arbi 
trarily  chosen  curve,  whether  plane  or  twisted  (as  long  as  we  avoid  singular 
points  of  the  equation).  See  Goursat-Bourlet,  p.  21. 

2°  Consider  the  equation/ (x,y,  z, p,  q,  r,  s,  /)  =  o.  If  this  contains  neither 
r  nor  /,  a  linear  transformation  will  introduce  one  or  both  of  these.  We  shall 


*  Up  is  absent,  then  q  must  appear,  and  the  argument  here  employed  can  be  car 
ried  out  by  interchanging  in  it/  and  q,  and  x  and^y. 

f  The  function  F(x,ytz,q)  is  said  to  be  regular  in  the  regions  of  x  =  x0,y=yn, 
z  =  z0,  q  —  qQ,  if  it  can  be  developed  by  Taylor's  theorem  in  a  convergent  power  series 
in  x  —  x0,y  —  y0t  z  —  z^q  —  q0. 

t  By  a  singular  point  of  the  equation  we  mean  one  whose  coordinates  together 
with  the  corresponding  value  of  q  determine  a  set  of  values  in  the  regions  of  which  P 
ceases  to  be  regular. 


204  DIFFERENTIAL   EQUATIONS  §78 

suppose  then  that  one  of  them  appears ;   and  there  will  be  no  loss  in  supposing 
that  it  is  r.     Solving  for  this,  the  equation  takes  the  form 

r  =  F(xty,  z,p,  q,  s,  /). 

The  existence  theorem  tells  us  that  if  F  is  regular  in  the  regions  ofx  =  #0, 
y  —  yo>  z  =  20 ,p=po,q  =  qQ>s  =  SQ,t  =  /0,  and  if  <f>( y)  and  f(y)  are  any 
arbitrarily  chosen  functions  of  y,  regular  in  the  region  ofyo,  and  such  that  <t>(yo) 
=  20,  0'Oo)  =  ?o,  0"Oo)  =  A),  ^Oo)  =  /o,  t'(yo)  =  so,  there  exists  one  and 
only  one  function,  z,  ofx  and  y  which  is  regular  in  the  regions  ofx  =  XQ,y  —  y^ 
and  such  that  z  =  0(j)  and p  =  ^(jv)  for  x  =  XQ. 

Looked  at  geometrically,  this  means  that  given  any  curve  z  =  <t>(y)  in  the 
plane  x  =  XQ,  there  exists  an  indefinite  number  of  integral  surfaces  passing 
through  it.  But  if  at  each  point  of  this  curve  we  fix  a  tangent  plane,  there  exists 
one  and  only  one  integral  surface  through  the  curve  and  tangent  to  these  planes, 
For,  the  direction  cosines  of  the  normal  to  the  tangent  plane  are  proportional 
to/,  q,  —  i.  As  soon  as  the  curve  is  given  we  know0(j).  The  q  at  each  point 
of  this  curve  is  0'(jy),  hence  it  is  also  known.  So  that  to  give  the  tangent  plane 
at  each  point  is  to  give/,  which  is  our  ^(j).  Once  0(j)  and  ^(jv)  are  given, 
the  existence  theorem  says  the  integral  surface  is  determined  uniquely. 

As  in  the  previous  case,  this  may  be  extended  to  apply  to  any  curve  whether 
plane  or  twisted. 

Here  again,  no  singular  points  of  the  differential  equation  are  supposed  to 
appear  in  the  regions  in  which  we  are  interested. 


CHAPTER   XIII 
PARTIAL  DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER 

79.  Linear  Partial  Differential  Equations  of  the  First  Order. 
Method  of  Lagrange.  —  Lagrange  deduced  the  following  very  neat 
method  of  solving  linear  partial  differential  equations  of  the  first 
order.  The  general  type  of  such  an  equation  for  one  dependent  and 
two  independent  variables  is 


(l) 

where  P,  Q,  R  are  functions  of  x,  y,  z. 

Consider  the  linear  equation  with  three  independent  variables  x,  y,  z, 


which  is  homogeneous  (i.e.  there  is  no  right-hand  member),  and  the 
coefficients  are  functions  of  the  independent  variables  only. 

If  u  —  c  satisfies  (i),  u  will  be  a  solution  of  (2)  ;  for,  differentiating, 


,                                dx  dy 

we  have            £  =  —  —  ,  #==——• 

J?*  du 

dz  dz 

Substituting  these  in  (i)  we  get  (2). 

205 


206  DIFFERENTIAL   EQUATIONS  §  79 

Conversely,  if  u  is  a  solution  of  (2),  u  =  c  will  satisfy  (i).     For, 
from  (2)  we  have,  on  solving  for  7?, 


dx 

But  ~ 


dz 


when  u  =  c.     Hence  (i)  follows.     From  this  we  see  that  the  problem 
of  solving  (i)  is  equivalent  to  that  of  solving  (2). 

Consider  now  the  system  of  ordinary  differential  equations, 

,  ^  dx  _dy  _  dz 

~P~~Q~ll' 

If  u  is  a  solution  of  (2),  u  —  c  satisfies  (3)  ;  for,  if  we  multiply 

numerator  and  denominator  of  these  fractions  by  —  ,  —  ,  —  respec- 

dx    dy     dz 

tively,  we  have,  by  composition,  that  each  of  the  fractions  of  (3)  is 

equal  to 

du   ,    ,  ou   ,    ,  du  j 


dx  dy  dz 

Since  the  denominator  is  zero  by  hypothesis,  the  numerator  (which 
is  du)  must  be.  Hence  u  —  c  is  a  solution  of  (3)  [§  65,  3°,  (£)]. 

Conversely,  if  u  =  c  is  a  solution  of  (3),.  u  will  satisfy  (2).  For,  by 
differentiation  we  have 

du  j    .  du   , 


79  PARTIAL,   OF  THE  FIRST  ORDER  2O? 

To  say  that  u  =  c  satisfies  (3)  is  to  say  that  for  it 
dx:dy\dz  =  P\  Q  :  R. 

Hence,  replacing  in  (4)  the  former  by  the  latter,  we  have 


++= 

dx  dy  dz 

which  shows  that  (2)  is  satisfied. 

Hence  the  problem  of  solving  (2)  is  reduced  to  that  of  solving  (3). 

Therefore,  finally,  the  problem  of  solving  (i)  is  reduced  to  that  of 
solving  (3),  since  every  solution  of  the  latter  is  also  a  solution  of  the 
former,  and  conversely. 

Moreover,  if  u=  cly  v=  c2  are  any  two  independent  solutions  of 
(3),  any  function  of  u  and  v,  say  •<£(&,  v),  will  satisfy  (2).  For, 
substituting  this  in  the  left-hand  member  of  (2),  we  get 


Q  +  +  Q+* 

du  \     dx          dy          dz  J      dv\     dx          dy          dz 

This  vanishes,  since  u  and  v  each  satisfy  (2).  Hence  <j>(u,  v)  is 
also  a  solution  of  (2)  irrespective  of  the  choice  of  the  function  <£. 

Therefore,  <j>(u,  v)  =  o*  is  the  general  solution  of  (i),  since  it  con 
tains  an  arbitrary  function,  §  78.  Since  </>  is  an  arbitrary  function, 
there  is  no  loss  in  putting  zero  for  the  right-hand  member  instead 
of  an  arbitrary  constant. 

Word  for  word,  the  above  proof  may  be  extended  to  a  linear  equa 
tion  with  n  independent  variables.  So  that  we  can  formulate  the  rule  : 

To  find  the  general  solution  of 


D  i      D 

"i-z  --  h  /17  --  1  » 

dxl  dx2  dxn 

dx\      fix*  dxn      dz 

^=^=-  =  7C  =  T 

The  solution  may  obviously  also  be  written  in  the  form  u  =/(v),  OTV= 


2O8  DIFFERENTIAL  EQUATIONS  §79 

If  the  general  solution  of  this  system  is 


then   <j>(u1}   u2,   •••,   un)  =  o,   where  $  is  an   arbitrary  function  of 
ui>  U2>  '">  UM  will  be  the  general  solution  of  '(i). 

Ex.  1.    xzp  +  yzq  =  xy. 

dx  _dy  _dz 

xz     yz     xy 

Multiplying  numerator  and  denominator  of  the  three  members  by 
y,  x,  —  2Z  respectively,  we  have,  by  composition,  since  the  denomi 
nator  vanishes, 

ydx  -\-xdy-  2zdz  =  o.     (Method  3°,  (£),  §  65.) 


Besides,  =      ,  or        =       gives      =  cz. 

xz     yz          x       y  x 

.:  <f>  (xy  —  z2,   2-  j=  o  is  the  general  solution. 


Ex.2.    _         +         + 

dx        dy 

dx  _dy  _     dz     _du 
—  y      x       i  +  z2      o 

We  have  at  once  xdx  +ydy  =  o, 

or 
Also, 


§80  PARTIAL,    OF  THE   FIRST   ORDER  209 

Multiplying  numerator  and  denominator  of  the  first  two  members 
by  —  y  and  x  respectively,  we  have  by  composition, 


(Method  3°,  («),  §  65.) 


is  the  gen- 

eral  solution. 

Ex.  3.  yp  —  xq  =  x2  —y2. 
Ex.4.         —  z 


80.  Integrating  Factors  of  the  Ordinary  Differential  Equation 
Ndy  =  o.  —  We  are  now  in  a  position  to  treat  satisfactorily 
the  problem  of  finding  an  integrating  factor  for  an  ordinary  differ 
ential  equation  of  the  first  order  and  degree.  We  have  seen  (§  7) 
that  the  necessary  and  sufficient  condition  that  /x  be  an  integrating 
factor  of  Mdx  +  Ndy  =  o  is 


dx  By 


or  /t  —  -  —     +  N?£ -M^  =  o,  whence 

\dx       By  J          Bx          By 

JV         Bu,  M         da 


*  Since  the  number  of  solutions  of  this  linear  partial  differential  equation  is  infinite, 
we  see  again  that  an  ordinary  differential  equation  of  the  first  order  has  an  infinite 
number  of  integrating  factors  (§  5). 


2IO  DIFFERENTIAL  EQUATIONS  §80 

To   find   /x,   satisfying   (i)    we   consider   the  system  of  ordinary 
equations, 


(  N  dy        dx  dx 

^-*- 


Remark.  —  In  actual  practice,  when  trying  to  integrate  M  dx  +  Ndy  —  o,  we 
are  not  desirous  of  finding  the  most  general  form  for  /x  ;  as  a  matter  of  fact,  as 
a  rule,  the  simpler  the  form  the  better.  Hence  any  one  solution  of  (2)  will  be 
sufficient.  It  should  be  noted  that  this  is  not  usually  a  practical  method.  But, 
for  special  forms  of  M  and  N,  a  solution  of  (2)  can  be  found.  Thus  the  follow 
ing  general  classes  of  equations  for  which  we  can  find  a  solution  of  (2)  may  be 
noted  here.* 


i°  If  -Z  -  -  is  a  function  of  x  only,  say^(jf),  we  have,  from 

(2),  that  p.  =  <?Jyi(z)d*  is  an  integrating  factor  (§  17). 
BN     dM 

2°  If  —  _  2>-  is  a  function  of  y  only,  say/2(j),  then^=^^^  is 

obviously  an  integrating  factor  (§  17). 

3°  If  the  equation  is  linear,  then  M=  Py  —  Q,  N=  i,  and  (2) 

becomes  Pdx  =  —  —  -  dy  =  -^  •     Hence  //,  =  efpdx  is  an  integrating 
factor  (8x3).     ^~Q'        * 

4°  If  M  and  N  are  homogeneous  and  of  the  same  degree  n,  we 
get,  by  composition,  after  we  have  multiplied  numerator  and  denomi 
nator  of  the  first  two  members  of  (2)  by  y  and  x  respectively, 

/    BM       BN\,    ,   /    BN         BM\, 
(  y  --  y  -  }dx  +  [  x  --  x  -    dy 

/-N  \  *y       a*)       \   d*       ty  i    =^ 

xM+  yN  p. 

*  These  were  enumerated  in  Chapter  II,  in  the  list  of  equations  for  which  integrat 
ing  factors  can  be  found. 


§81  PARTIAL,   OF  THE   FIRST  ORDER  211 

By  Euler's  theorem  for  homogeneous  functions,  we  have 


dM, 

dM        .. 

dM 

a^4 

y~fy  =       " 

'     -'-y  dy 

cdN+ 

v*N-nN 

5N 

c^+ 

•y-fy-n" 

'    '''XHx 

—-, 
dx 

dN 

-—-t 

dy 


whence  (3)  may  be  written 


-  n 


xM+yN  ~     —' 

But  Mdx  +  N  dy  =  o.     Hence  we  may  add  (n  +  i)(Mdx  +  Ndy) 
to  the  numerator  on  the  left  without  altering  its  value.     Doing  so,  we 

d(xM+yN)=      dp 
xM+yN  /«. 

Integrating,  we  have  i          /s  T^N 


5°  If  M=yfl(xy)9  N=  xf^xy),  then  on  multiplying  numerator  and 
denominator  of  the  first  and  second  members  of  (2)  by  y  and  —  x 
respectively,  we  have  by  composition,  after  obvious  reductions, 


xM—yN         p 

On  integrating,  we  have  i          ,g      . 

'7 


81.    Non-linear  Partial   Differential    Equations    of    First    Order. 
Complete,  General,  Singular  Solutions.  —  We  have  seen  (§  76)  that 
a  primitive 
(i)  4>(x,y,  z,  a,b}  =  Q 


212 


DIFFERENTIAL  EQUATIONS 


§81 


which  involves  two  arbitrary  constants  gives  rise  to  a  unique  partial 
differential  equation  of  the  first  order 


This  differential  equation  is  gotten  by  eliminating  a  and  b  from  (i) 
and  its  derivatives  with  respect  to  x  and  y  respectively ;  viz., 


(3) 


> 
dx          dz 

d<f>  .      d<f> 

s    +? s =a 
dy          dz 


(i)  is  spoken  of  as  a  solution  of  (2).  Here,  of  course,  a  and  b  are 
constants.  Letting  them  be  parameters,  we  have,  on  differentiating 
(i),  and  taking  account  of  (3), 


(4) 


3<£  da      d(j>  db  _ 
da  dx       db  dx 


d(f>  da  .__ 
da  dy       db  dy 


These  two  equations  can  be  consistent  only  in  case  either 


(5) 


da 


or 


da  db_ 

dx  dx 

da  d^ 

dy  dy 


=  o. 


§8i  PARTIAL,   OF  THE   FIRST   ORDER  213 

If  this  determinant  vanishes,  b  is  some  function  of  a,  say  ^(a),  (Note 
I  in  the  Appendix).  Then  (4)  may  be  replaced  by 

(6) 

Since  (5)  and  (6)  were  gotten  on  the  assumption  that  (3)  hold,  it 
follows  that  if  we  eliminate  a  and  b  from  (i)  and  (5),  or  from  (i) 
and  (6),  we  shall  get  relations  between  x,  y,  z  which  will  satisfy  (2). 
Hence  these  relations  are  also  solutions. 

We  see  then  that  the  primitive  of  a  partial  differential  equation  of 
the  first  order  is  not  the  only  solution.  But  since  the  others  can  be 
gotten  from  it,  Lagrange  called  it  the  complete  solution. 

We  have  already  noted  (§  78)  that,  in  the  general  theory  of  partial 
differential  equations,  it  is  proved  that  an  arbitrary  function  appears 
in  the  general  solution  of  an  equation  of  the  first  order.  Since  in  (6) 
the  function  \\i(a]  is  any  function  of  a  we  please,  Lagrange  called  the 
solution  gotten  by  eliminating  a  and  b  from  (i)  and  (6)  the  general 
solution. 

A  particular  solution  is  gotten  by  assigning  definite  values  to  a 
and  b  in  the  complete  solution,  or  by  using  a  definite  function  ty(a) 
and  eliminating  a  and  b  from  (i)  and  (6). 

On  the  other  hand,  the  solution  gotten  by  eliminating  a  and  b 
from  (i)  and  (5)  contains  nothing  arbitrary,  and  is  known  as  the 
singular  solution.  It  is  the  exact  analogue  of  the  singular  solution 
of  ordinary  differential  equations.  Looked  upon  geometrically,  it  is 
the  equation  of  the  envelope  of  the  doubly  infinite  number  of  surfaces 
whose  equation  is  given  by  (i),  just  as  the  general  solution  is  the 
envelope  of  an  arbitrarily  chosen  single  infinity  of  those  surfaces. 
It  can  also  be  shown  *  that  the  singular  solution  can  be  gotten  from 
the  differential  equation  by  eliminating  p  and  q  from 

f(x,  y,  z,J>,2)=o,-~-  =  o,-f  =  o, 
op  oq 

*  See  Goursat-Bourlet,  p.  24,  also  p.  199  and  foil. 


214  DIFFERENTIAL   EQUATIONS  §81 

in  exact  analogy  to  the  case  of  ordinary  differential  equations  of  the 
first  order.  But  the  limits  of  this  book  prohibit  a  further  discussion 
of  the  subject.  We  shall  conclude  it  with  the  following  obvious 
remarks  : 

i°  There  need  be  no  singular  solution.  This  will  happen  in  case 
the  equations  (i)  and  (5)  are  inconsistent;  or,  geometrically,  in 
case  the  surfaces  (i)  have  no  envelope. 

2°  The  general  solution  cannot  be  written  down.  ij/(a)  is  entirely 
at  our  disposal  ;  but  until  it  is  given,  there  is  no  way  of  eliminating 
a  and  b  from  (i)  and  (6),  and  when  it  is  given,  we  have,  of  course,  a 
particular  solution. 

3°  There  is  no  unique  complete  solution.  Any  solution  involving 
two  arbitrary  constants  may  be  taken  as  one.  It  is  easy  to  see  that 
there  is  an  indefinite  number  of  them.  For,  if  we  choose  any  form 
for  \l/(a)  which  involves  two  arbitrary  constants  h  and  k,  on  eliminat 
ing  a  and  b  from  (i)  and  (6)  we  get  a  solution  involving  these  two 
arbitrary  constants,  which  fulfills  all  the  requirements  for  a  com 
plete  solution. 

We  saw  in  (§  76)  that  a  complete  solution  of  z2(/2  +  q*  +  i)  =  P?  is 
O  -  *)2  +  (y 


which  represents  a  family  of  spheres  of  radius  R  and  centers  in  the  plane  2=0. 
The  envelope  of  these  is  the  pair  of  planes  z2  =  I&,  which  is  obviously  the  result 
of  eliminating  a  and  b  from 


or  of  eliminating  p  and  q  from 

22  (/2  +  ?2  +  I)  =  &,  z2/  =  O,  z*q  =  O. 

z2  =  K1  is  then  a  singular  solution. 

The  choice  of  any  relation  b  —  \fy  (a)  results  in  selecting  those  spheres  whose 
centers  lie  along  the  curve  y  —  $(x}  in  the  plane  z  =  o.  The  envelope  of  these 
is  obviously  the  tubular  surface  traced  out  by  the  motion  of  a  sphere  of  radius  R 


§82  PARTIAL,   OF  THE   FIRST  ORDER  215 

moving  with  its  center  on  the  curve  y  —  ^(x)  and  along  its  entire  length.     In 
particular,  if  ^(jf)  is  a  linear  function  of  x,  the  envelope  is  a  cylinder. 

As  an  example,  suppose  b  —  ha  +  k. 

To  find  the  corresponding  solution,  we  have  to  eliminate  a  from 


(x  -  a)2  +  (y  -  ha  - 

O  -  a)  +  h(y  -  ha  -  k}  =  o. 

From  the  second  of  these  we  have 

_x  +  ky-hk 

i+h* 

Putting  this  in  the  other  equation,  we  have 

(hx  -y  +  W  =  J?1      3 


(I+/&2)2 
or  (hx  -  y  +  ky  =  (i  + 

which  is  obviously  the  equation  of  a   circular  cylinder  whose  axis  is  the  line 

y  =  hx  +  k,  z  —  o. 

Since  this  solution  contains  two  arbitrary  constants,  it  is  a  complete  solution. 
(See  Ex.  6,  §  83.) 

As  an  exercise,  the  student  should  start  with 

(hx-y+  X)*  =  (I  +  #)(*«  -  32) 

as  a  primitive  and  show  that,  considering  h  and  k  as  the  arbitrary  constants,  the 
resulting  partial  differential  equation  is  again  z2(/2  +  qz  +  i)  =  HP. 

82.  Method  of  Lagrange  and  Charpit.  —  Since  the  knowledge  of  a 
complete  solution  is  sufficient  to  enable  us  to  find  all  other  solutions, 
it  is  usually  desirable,  whenever  possible,  to  find  it  first.  Lagrange 
suggested  the  following  method  : 

Given  the  equation 
(i)  f(*,y,  *,/,?)  =<>* 


2l6  DIFFERENTIAL   EQUATIONS  §82 

try  to  find  a  second  relation 

(2)  4>(x,y,z,p,q,a)=o, 

which  involves  either  p  or  q  or  both,  and  also  an  arbitrary  constant, 
and  such  that  when  we  solve  (i)  and  (2)  for/  and  q  and  put  these 
in  the  total  differential  equation 

(3)  dz=pdx+qdy 

the  latter  can  be  solved  ;  the  solution  of  (3)  is  a  complete  solution 
of  (i),  since,  from  what  precedes,  it  determines  z  as  such  a  function 
of  x  and  y,  that  the  values  of  /  and  q  obtained  from  it,  together  with 
z  satisfy  (i)  ;  besides  it  involves  two  arbitrary  constants.  To  do  this 
we  may  proceed  as  follows  : 

Differentiating  (i)  and  (2)  with  respect  to  x,  we  have 


(4)  +fi 

dx^Pdz      dpdx      dqdx 


dp  dx      dq  dx 
Similarly,  differentiating  with  respect  to  y, 


dy        dz      dpdy      dq  dy 

t~\  ?i  +  ^  +  ?**+**^=o 

By       ydz       dp  dy      dq  dy 
Now  the  condition  that  (3)  shall  be  integrable  is  obviously 

(8)  %-T  =  °- 

dy      dx 

*  Throughout  this  section,   -f  =  -$-  +  p^-,   ^-  =  ~-  +  q  ~- 
dx      dx         dz    dy      dy          dz 


§82  PARTIAL,    OF  THE   FIRST   ORDER  2  1/ 

From  equations  (4),  (5),  (6),  (7),  (8)  we  can  eliminate  the  four 

quantities  ~£,  -£y  -£,  -£  •    We  get  rid  of  -f    by  multiplying  (4) 
dx    dy    dx    dy  dx 

and  (5)  by  -^  and  -^  respectively,  and  subtracting.     The  result  is 
op  op 


dx        dzjdp      \dx        dz     dp        dq  dp       dq   dpjdx 


Multiplying  (6)  and  (7)  by  —  and  J-  respectively,  and  subtract- 

oq  dq 

ing,  we  get  rid  of  -^  and  have 
dy 


4.__L.         -        - 

dy         dzjdq      \dy       ^dzjdq      \dpdq       dp  dq)  dy 


Making  use  of  (8),  we  have,  on  adding  and  arranging  according  to 
derivatives  of  <£, 


dx        dzjdp      \dy        dzdq       dp  dx      dq  dy 


»_z_  _^_  q^L.    _r  =0. 
dp         dqj  dz 


This  is  a  linear  equation.     Hence,  to  solve  for  <£,  we  consider  the 
system  of  ordinary  equations 

(12)          dp  dq  dx  dy  dz  d$ 


_+ 

ft*        «b       fy        to  dp  dq          \Sp 


2l8  DIFFERENTIAL   EQUATIONS  §82 

Our  aim  is  to  find,  not  the  most  general  form  of  <£,*  but  any  form 
of  it  that  contains  p  or  q,  and  an  arbitrary  constant.  Hence,  in  actual 
practice,  we  look  for  the  simplest  one. 

Remark.  —  It  is  desirable  that  (11)  or  (12)  be  committed  to  memory. 

V  we  pot  |=|_+^l,and^=|-  +  ?J, 

ax      dx        dz  dy      Qy         Qz 

we  have  the  skew  symmetrical  form 

,  dp_      dx_dq_      dy, 

£~~3?~-#~~F 

dx          dp      dy          dq 

as  for  the  next  term ,  this  is  really  a  result  of  the  equality  of 


— —=      y      as  may  be  seen  by  composition  after  multiplying  numerator  and 

dp          dq 

denominator  of  the  first  by/  and  of  the  second  by  g,  and  taking  account  of  (3). 
Or  using  the  same  notation  as  before,  and  writing 


_ 
dx  dp       dp  dx      U> 


(u)  may  be  put  in  the  compact  form 

(»')  [/*]*,,+  [/*],,,  =  a 


*  Lagrange  thought  originally  that  by  using  the  general  form  of  0  (which  would  in 
volve  an  arbitrary  function)  he  could  get  the  general  solution  by  this  method.  But  this 
is,  in  fact,  not  practicable.  Charpit,  in  a  memoir  presented  to  the  Academie  des 
Sciences,  in  June,  1784,  suggested  the  use  of  any  form  for  0  involving  p  or  q  and  an 
arbitrary  constant,  obtaining  by  its  use  the  complete  instead  of  the  general  solution. 


§82  PARTIAL,  OF  THE  FIRST  ORDER  2IQ 

Ex.  1      z—pq=  o. 
To  find  <£  we  must  find  a  solution  of 

dp^  _<fy  _dx__dy__t^ 
p        q        q       p 

Using  the  first  two  members,  we  have 

$=p  —  aq  =  o. 

Combining  this  with  the  original  equation,  we  have 


Then  (3)  becomes  dz  —  a  \-  dx  -f-  \/_  dy, 

*  a  '  a 


Integrating,  we  have    2  Vtfz  =  ax  -f-  y  +  b, 
or  4  a2=  (tfjf  +y  +  ^)2, 

which  is  a  complete  solution. 

If  we  had  used  the  second  and  third  members,  we  would  have  had 

<f>  =  q  —  x  —  a  =  o.     Whence 

q  —  x  -f  a,  and  /  = 


x+  a 

Then  (3)  becomes  dz  =  dx  -\-  (x  +  a)  dy, 

x  +  a 

or  dv  =  (* +<*><&- 


22O  DIFFERENTIAL   EQUATIONS  §82 

Integrating,  we  have  y  +  b  =  -     —  , 

or  z  —  (x  +  d)(y  +  £), 

which  is  also  a  complete  solution. 

The  general  and  singular  solutions  can  be  gotten  from  either  of 
these.  Thus,  using  the  second  one,  we  get  the  general  solution  by 
eliminating  a  from 


and  y  +  $(a)  +  $\a)(x  +  a)  =  o, 

where  1/^(0)  is  any  function  of  a. 

In  particular  let  the  student  show  that  if  we  put 
^(«)  =  k  -  ha 

where  h  and  k  are  any  constants,  the  corresponding  solution  is  $hz=  (hx  -\-  y  -\-  £)*, 
which  we  recognize  as  the  first  form  obtained. 

The  singular  solution,  resulting  from  eliminating  a  and  b  from 

z=(x  +  a)(y  +  t), 
y  +  b  =  o, 

x  +  a  =  o, 
is  z  =  o. 

Let  the  student  show  that  this  can  also  be  gotten  from  the  other  form  of 
solution. 

Ex.  2.  /  =  (z  +yq)*. 
Ex.  3.    V/  -f  V^  =  2  x. 


§83  PARTIAL,  OF  THE   FIRST   ORDER  221 

83.  Special  Methods.  —  Special  methods  for  certain  forms  of  the 
differential  equation  at  times  prove  simpler  than  the  general  method 
of  §  82  (although  most  of  them  are  suggested  by  the  latter).  Some 
of  these  are  the  following  : 

i°   Suppose  all  the  variables  absent.     The  equation  takes  the  form 

(1)  f(P,  ?)  =  o, 
and  the  equations  for  determining  <£  become 

(2)  f£=^=.... 

O  O 

From  the  first  member  we  see  that  p  =  a. 

Then  q  is  gotten  by  substituting  in  (i).  Evidently  q  =  b,  where 
b  is  determined  by/  (a,  b)  =  o. 

We  have  then  dz  =  adx  -f-  bdy, 


or  z=ax  +  fy  +  c,  where  /  (a,  b)  =  o. 

Hence  the  rule  : 
The  complete  solution  of  f(p,  q)  =  o  is 

z  =  ax  +  by  +  c,  where  /(a,  b)  =  o.* 

By  means  of  simple  transformations,  certain  forms  of  equations  can  be  brought 
into  this  type  : 

Putting  log  s  =  Z,  orz=*Z,  wehave/=  ^  =  tfzM  =  2^?;   so  that 

dx  fix        o* 


Similarly  £  =        . 

*      By 

*  The  complete  solution  represents  a  doubly  infinite  set  of  planes.  Any  particular 
solution  represents  the  envelope  of  a  chosen  single  infinity  of  these.  This  is  a  develop 
able  surface.  There  is  no  singular  solution.  Why  ? 


222  DIFFERENTIAL   EQUATIONS  §83 

Hence,  if  the  equation  is/  /£  ,  %  )  =  o,  log  z  =  Z  will  transform  it  i 
\*     zl 


into 


Again,  if  we  let  log  x  =  X,  or  x  —  exy 


m 
dx     dXdx      xdX  dX 


Similarly,  putting  logy  =  Y,  we  get  yq=   ~  • 

a* 

Hence 


f(xp,  q)  =  o  is  transformed  into  /  (  —  ,  —  J  =  o  by  log  x  =  X\ 

f(p,yq}  =  o  is  transformed  into/(  ^,  -SI-  )  =  o  by  logy  =  F; 

\dx    BY  I 

f(xp,  yq)  =  o  is  transformed  into/f  —  ,  —  2  )  =  o  by  log*  =  X,  \ogy  =  F; 

\dX    o  Y  ] 

f(*£t  1\  ^  o  is  transformed  into/  f  M.,  MA  =obyloga:  =  ^;  logz=Z; 
J  \  z      z]  \dX    dy  I 


and  so  on. 

Ex.  1.    /?  =  I. 

The  complete  solution  is  z  =  aw  +  by  +  r,  where  ab  —  i, 

or  2=«^  +  - 

a 


Ex.2.    g= 

Writing   this    in   the  form  £=i  +  —  ,  we  see  that  the  substitu- 

2;  z 

tion  Jf  =  log  ^,  Z=log  0,  will  transform  this  equation  into 
dZ  dZ 


§83  PARTIAL,   OF  THE  FIRST  ORDER  223 

The  complete  solution  is  Z=  aX+  (i  +  a)y  -f  c.     Passing  back  to 
the  original  variables,  we  have   log  z  —  a  log  x+  (i  +a)y-{-c. 

or 


Ex.  3.  px  -h  qy  —  I. 

Ex.  4.   x2p2  +fq*  =  £*. 

Ex.5.  yq=p. 

2°   If  x  and  7  are  absent,  the  equation  takes  the  form 

and  we  have  for  determining  <£ 

di)      do      i 
.'.  -£.  =  -*•,  whence  ^  =  ap. 

Substituting  in  (i),  we  have  f(z,  p,  ap)  =  o,  whence  /  =  i[/(z,  a\ 

and  dz  =p  dx  +  q  dy  becomes  =  dx  +  a  dy, 

\l/(z,  a) 

where  the  variables  are  separated. 

To  put  this  rule  in  shape  easily  to  be  carried  in  mind,  we  note  that, 
to  say  q  —  ap  is  to  say  that  z  is  a  function  of  x  +  ay,  by  the  general 
method  of  §  79.  If  we  put  x t+  ay  =  /,  we  have 

^  —  Q?L  —  —      _^?_     fjk 

and  the  equation  (i)  becomes  the  ordinary  differential  equation 

dz 


224  DIFFERENTIAL   EQUATIONS  §83 

in  which  the  independent  variable  is  absent.      Hence  the  variables 

can  be  separated  immediately  after  solving  for  —  .     We  have,  then, 
i         i  "t 

the  rule  : 

If  the  equation  is  of  the  form  f(z,  p,  q)  —  o,  put  x  -f-  ay  =  /,  which 

will  replace  p  by  —  ,  and  q  by  a  —  .     Put  these  values  in  the  equation 
dt  dt 

and  solve  for  — 
dt 

If  the  equation  has  one  of  the  forms 

/O,  xp,  q}  =  O,  /(z,  p,  yq)  =  O,  /  (z,  xp,  yq)  =  O, 

one,   or  both  of  the  substitutions  log  x  =  X,  log  y  =  Y  will  reduce   it  to  the 
above  form. 


Ex.  6. 

Putting  x  +  ay  =  f,  p  =  —  ,  q  —  a—  ,  and  the  equation  becomes 
dt  dt 


[(S)WoV.i-« 


Vi  +  a^ 


or  (i  +  az)(X2  -z2)  =  (x  +  ay  +  b}\ 

Ex.  7.   xp(\  +?)  =  qz. 
Ex.  8.    z  =  pq. 

3°  If  the  dependent  variable  is  absent,  and  the  equation  is  such 
that  it  can  be  put  in  the  form 

(0  fi(x.f) 


§83  PARTIAL,   OF  THE   FIRST   ORDER  22$ 

(a  sort  of  separation  of  the  variables),  the  equations  to  determine  <£ 
become 

0        *  =  ...*_.... 

^Ll  —^ 

dx  dp 

.\  -£  dx  +  —  dp  =  o,  or  df-L  =  o.     Hence  we  have 

dx  dp 


Solving  these,  we  have  p  =  \l/l(x,  a),  q  =  ^^(yy  a), 
and  dz  =p  dx  -\-  q  dy  becomes 

dz  =  $i(x,  a)dx  +  fa(y,  a)dy, 

in  which  the  variables  are  separated.     Hence  the  rule  : 

If  the  dependent  variable  is  absent,  and  the  other  variables  are 
separable,  such  that  the  equation  takes  the  form  fi(x,  p)  =^2(y}  q), 
equate  each  of  these  members  to  a  constant,  solve  the  resulting  equa 
tions  for  p  and  q,  and  put  these  values  in  dz  =p  dx  +  q  dy. 

If  the  equation  can  be  put  in  the  formyif  x,  £\=f2(y,  i  j,  the  transforma 
tion  log  z  =  Z  will  reduce  it  to  the  form  above. 

Ex.  9.    q—  2ypz. 

Ex.10.    2(zx  —  zy]  —  p  +  q  =  o. 

4°  The  equation  z  —px  -f  qy  -f-/(/,  q),  which  is  usually  referred  to 
as  the  extended  Clairaut  equation  (§  27),  will  obviously  be  solved  if 
we  put  /  =  a,  q  =  b.  We  have  then  the  complete  solution 

z  —  ax  -f-  by  +f(a,  b). 


226  DIFFERENTIAL   EQUATIONS  §83 

While  the  general  method  of  §  82  applies  here,  it  does  not  give  this  simple 
form  of  solution.  By  that  method  we  may  use  either  p  =  a  or  q  =  b,  but  not  both 
simultaneously.  As  a  matter  of  fact  it  is  an  accident  if  the  result  of  substituting 
in  the  differential  equation  the  values  of  p  and  q  obtained  from  two  solutions  of 
equations  (12),  §  82  is  a  complete  solution.  It  does  happen,  at  times,  as  in  the 
case  in  question.  But  there  is  no  certainty  that  it  will,  nor  is  there  even  a  likeli 
hood  of  it. 


Ex.  11.    Solve  z  =px  +  qy  +  V/2  +  q2  +  i,  and  examine  for  singu 
lar  solution. 

5°  If  the  equation  is  of  the  formf(x+y,p,  ^)  =  o,  let  q= 
Then/O  +y,  p,f  +  a)=o  gives  /  =  <j>(x  +y,  a),  whence 


and  the  equation  dz  =  p  dx  -f-  q  dy  becomes 

dz  =  <j>(x  +y,  a)(dx  +  dy)  +  a  dy. 

Let  the  student  show  that  this  form  of  solution  is  given  by  the  general  method 
of  §  82. 

If  the  equation  is  of  the  form/f  x  +  y,  *,  -2  J  =  o,  the  transformation  \ogz=Z 
will  reduce  it  to  the  form  above. 

Ex.  12.  p(x+y)  —  ?  =  o. 

Ex.  13.    zp(x  +y)  +p(q  -/)  =  z2. 

6°  If  either  p  or  q  is  absent,  the  method  of  solution  is  obvious 
by  inspection.  In  the  former  case  integrate  considering  x  as  a  con 
stant,  when  the  constant  of  integration  will  be  an  arbitrary  function 
of  x.  In  the  other  case  integrate  considering^  as  a  constant,  when 
the  constant  of  integration  will  be  an  arbitrary  function  of  y.  These 
solutions,  involving  arbitrary  functions,  are  general  solutions.* 

*  When  these  equations  are  of  the  first  degree  in  the  derivative,  they  are  linear 
equations.  The  method  here  given  is  exactly  that  of  Lagrange  for  such  equations 
(§  79)- 


84  PARTIAL,   OF   THE   FIRST   ORDER  22? 

Ex.  14.    (x  —y}q  —  (x  +  z)  =  o. 

Considering  x  as  a  constant,  we  can  write  q  =  —  • 

dy 

dz          dy 

—  =  o. 


x+z     x—y 

Integrating  and  taking  exponentials  of  both  sides,  we  have 

(x  +  z)(x-y)  =  <t>(x), 
where  </>(V)  is  an  arbitrary  function  of  x. 


Ex.  15.    xp*  — 

Ex.16.  p+y(z— x)  =  o. 

Ex.17.   /(/-i)=*2/- 

84.  Summary.  — Partial  differential  equations  of  the  first  order  are 
divided  into  two  general  classes  :  those  which  are  linear  in  the 
derivatives  of  the  dependent  variable,  and  those  which  are  not. 

i°  For  the  solution  of  linear  differential  equations  of  the  first 
order  the  method  of  Lagrange  applies,  giving  the  general  solution 

(§  79)- 

2°  For  the  solution  of  non-linear  equations  of  the  first  order,  the 
general  method  of  Lagrange  and  Charpit  applies  (§82),  giving  a 
complete  solution.  From  this  the  other  solutions  can  be  gotten 
(§81). 

At  times  the  special  methods  of  §  83  are  shorter  than  the  general 
method  of  §  82. 

Sometimes  a  transformation  of  variables  will  help  in  the  solution 
of  an  equation. 


228  DIFFERENTIAL  EQUATIONS  §  &4 

EX.  i.  a 

EX.      2.          = 


~,  du   ,      du  ,     du 

Ex.    3.    x—+y—-+z  — 

ox         dy         oz 


Ex.  4.  z= 

Ex.  5.  xypq  =  z2. 

Ex.  6. 

Ex.  7.  q  = 

Ex.  8. 

Ex.  9.  (x+y)(p  +  qj+(x-y}(p-qf=i.       [Let  * 


Ex.  10. 

Ex.  11.  (x*  +  y2)  (pz  +  /)=i.     [Let  x  =  p  cos  0,  y  =  p  sin 

Ex.  12.  (/  +  02 - x*)p -2xyq+2xz=o. 

Ex.13.  q^  —  z^(p  —  (j}. 

Ex.  14.  (y  — 

Ex.15.  z  —  xp—yq=2 

Ex.  16.  pq  =px+  qy. 


Ex.17.     (j+ «+*)£+ (»_+•* 

5^  5y 


§  84  PARTIAL,  OF  THE   FIRST   ORDER 

Ex.  18.  Determine  a  system  of  surfaces  such  that  the  normal  at 
each  point  makes  a  constant  angle  with  the  plane  of  xy. 

Ex.  19.  Determine  a  system  of  surfaces  such  that  the  coordinates 
of  the  point  where  the  normal  meets  the  plane  oixy  are  proportional 
to  the  corresponding  coordinates  of  the  point  on  the  surface. 

Ex.  20.  Determine  a  system  of  surfaces  for  which  the  product  of 
the  distances  of  the  tangent  plane  from  two  fixed  points  is  a  constant. 


CHAPTER  XIV 

PARTIAL  DIFFERENTIAL  EQUATIONS  OF    HIGHER  ORDER 
THAN  THE  FIRST 

85.  Partial  Differential  Equations  of  the  Second  Order,  Linear  in 
the  Second  Derivatives.  Monge's  Method.  —  The  general  type  of  a 
partial  differential  equation  of  the  second  order  linear  in  the  second 
derivatives  is 


where  R,  S,  T,  V  are  functions  of  x,  y,  z,  p,  q.  Gaspard  Monge 
(1746-1818)  suggested  a  method,  which  is  known  by  his  name,  by 
which  a  first  or  intermediary  integral  is  found  in  the  form  Of  a  partial 
differential  equation  of  the  first  order  involving  an  arbitrary  function. 
The  solution  of  this  equation  by  any  of  the  methods  of  Chapter 
XIII,  or  otherwise,  will  then  give  the  general  solution.  While  this 
method  applies  only  in  case  R,  S,  T,  V  satisfy  certain  conditions,  it 
works  sufficiently  frequently  to  justify  our  giving  here  at  least  the 
rule  by  which  solutions  are  gotten  by  this  method.*  Besides 

(2)  dz=pdx  +  qdy, 

we  have 


—  sdx+tdy. 

Eliminating  r  and  /  from  (i)  and  (3)  we  have 
(4)  s(R  dy*  -Sdxdy  +  Tdx2)  -(Rdy  dp  +  Tdx  dq  -  V  dx  dy)=*  o, 

*  For  a  detailed  account  of  this  subject  see  Forsyth,  p.  358   and  foil.,  or  Boole, 
Chapter  XV. 

230 


§85  PARTIAL,   HIGHER  ORDER  THAN  THE  FIRST 

Whenever  it  is  possible  to  satisfy  simultaneously, 

(5)  Rdy^-Sdx  dy  +  7V*2  =  o, 

(6)  R  dy  dp  +  T  dx  dq  —  V  dx  dy  =  o* 

(4)  will  be  satisfied  and,  therefore,  so  will  (i).  (5)  is  equivalent  to 
two  equations  of  the  first  order, 

(7)  dy-  Wi(xiy,ztptq)dx  =  Q,<fy-Wi(x9ytz,p,q)dx  =  <>, 

which  become  identical  in  case 

(8)  4  *T=  S\ 

Equations  (2)  and  (6),  together  with  either  one  of  (7),  constitute 
a  system  of  three  total  differential  equations  in  the  five  variables  x,  y, 
2,  pt  q.  Such  a  system  can  be  solved  only  in  case  certain  conditions 
are  fulfilled,  and  it  is  for  this  reason  that  Monge's  method  does  not 
always  work.  It  will  work  if  we  can  find  two  independent  solutions 
of  this  system 

*i(#,  y,  z,  P,  ?)  =  £i>     Ui(x,  y,  z,  p,  q)  =  *,. 

In  this  case  it  turns  out  that 

(9)  *i  =  ^(^2), 

where  <^>  is  an  arbitrary  function,  is  an  intermediary  (or  intermediate) 
integral  Looked  upon  as  a  partial  differential  equation  of  the  first 
order,  (9)  must  be  integrated  again.  Its  general  solution  will  be  the 
solution  of  (i). 

In  case  it  happens  that  not  only  one  of  (7),  but  each  one,  together 
with  (2)  and  (6),  determines  a  system  that  can  be  solved,  we  have 

*  Equations  (5)  and  (6)  are  usually  referred  to  as  Monge's  equations. 


232  DIFFERENTIAL   EQUATIONS  §85 

two  intermediary  integrals  (9).  Solving  these  for  /  and  g,  we  put 
the  values  of  the  latter  in  dz  =p  dx  -f  q  dy.  The  integral  of  this  is 
the  general  solution  of  (i). 

Ex.  1.    fr  —  2  pqs  4-  p^t  —  o. 
Monge's  equations  are 


2pqdxdy  +/  Jx*  =  o, 
q^  dy  dp  -|-/2  dx  dq  =  o. 

The  first  of  these  is  a  perfect  square, 


Substituting  this  in  the  second  one,  it  becomes 
qdp-pdq=v, 

whence  «-  =  c^ 

q 

The  first  one,  combined  with  dz=p  dx  +  q  dy,  gives 

dfessO, 

whence  z  =  ^2. 

Hence  an  intermediary  integral  is 


In  this  case  we  have  only  one  intermediary  integral,  hence  we 
must  integrate  this.  Since  it  is  linear,  it  can  be  solved  by  the  method 
of  Lagrange  (§  79).  Its  general  solution  is 


5  PARTIAL,  HIGHER  ORDER  THAN  THE  FIRST  233 

Ex.  2.    r-a2t  =  o* 

Monge's  equations  are 

df  —  a  z  dx*  =  o,  or  dy  —  a  dx  =  o  and  dy  -f-  a  dx  =  o, 
dy  dp  —  a2  dx  dq  =  o. 
Using  dy  —  a  dx  =  o,  we  have  y  —  ax  =  c^. 

Combining  this  with  the  second  of  Monge's  equations,  we  get 
dp  —  a  dq  =  o  ;  whence  p  —  aq  —  c%. 

Hence  an  intermediary  integral  is 


Using  the  other  equation,  dy  +  a  dx  =  o,  we  get  a  second  interme 
diary  integral 


Solving  these  for  /  and  q,  we  have 


*  A  much  simpler  method  of  solution  for  this  equation  will  be  given  in  §  88.  This 
equation  plays  an  important  r&le  in  Mathematical  Physics.  It  was  first  integrated  by 
Jean-le-Rond  D'Alembert  (1717-1783)  in  a  memoir  entitled  Recherches  sur  les  vibrations 
des  cordes  sonores,  presented  in  1747  to  the  Berlin  Academy.  In  studying  the  vibra 

tions  of  a  stretched  elastic  string,  he  considered  the  equation  in  the  form  T^~V^~  o> 

where  /  is  the  time,  and  x  and  y  are  the  rectangular  coordinates  of  a  point  of  the  string, 
x  the  coordinate  measured  along  the  line  joining  the  extremities  of  the  string,  and  y  the 
displacement  of  the  point  from  the  position  of  equilibrium.  His  proof  is  given  in 
Marie,  Histoire  des  Sciences  Mathcmatiques  et  Physiques,  t.  VIII.  p.  217. 


234  DIFFERENTIAL  EQUATIONS  §85 

We  have  now  to  solve 
dz  =  I  l<t>(j  +  ax)  +  $(y  -  ax)}dx  +  ^  [<£(>  -f  ax]  -  \j,(y  -  ax)\dy 

—  -$(y  -f  ax)  (ay  +  a  dx)  -f-  -\j/(y  —  ax)  (ay  —  a  dx),  which  is  exact. 


Since  <£  and  \l/  are  symbols  of  arbitrary  functions,  we  shall  retain 
them  in  writing  the  solution 


z  =  <£(j  +  ax)  +  $(y  —  ax). 

There  is  no  loss  in  failing  to  add  an  arbitrary  constant,  since  either 
of  the  arbitrary  functions  may  be  supposed  to  incorporate  that. 


Ex.3.    r-/=- 

x+y 

Monge's  equations  are 

df  —  dx2  =  o,  or  ay  —  dx  =  o  and  dy  +  dx  =  o, 


dy  dp  —  dx  dq  -f  ~*    dx  dy  =  o. 

x+y 

Using  dy  —  dx  —  Q^  we  have  y  —  x  =  cl. 

Combining  this  with  the  second  of  Monge's  equations,  we  have 

2  x  dp  +  ^pdx  —  2  x  dq-\-  c^dp  —  dq)  =  o. 
Also  dz  =p  dx  +  q  dy  becomes 

dz  =  p  dx  -f-  q  dx. 

Subtracting  twice  this  from  the  above  equation,  we  get 
2(x  dp  +p  dx)  —  2(x  dq  +  q  dx)  -f  c^dp  —  dq)  +  2dz=.  o. 


§85  PARTIAL,  HIGHER  ORDER  THAN  THE   FIRST  235 

This  is  exact,  and  has  for  solution 


or  (x 

Hence  an  intermediary  integral  is 


Using  the  equation  dy  -f-  dx  =  o,  we  get  a  system  of  total  differen 
tial  equations  which  are  not  integrable.  Hence  we  must  integrate 
the  intermediary  integral.  This  is  linear,  so  Lagrange's  method 

applies, 

dx        —  dv  dz 


x+y     x+y 

From  the  equation  of  the  first  two  members  we  have 

x+y  =  a. 

Replacing  y  by  its  value  a  —  x,  we  have 
dx  dz 


a       <p(a  —  2  x)  —  2  z* 


or 


This  is  a  linear  ordinary  equation  of  the  first  order.     An  integra- 

-f         ?? 
ting  factor  (§  13)  is  e<*>dx  =  e^,  and  the  solution  is 

2x  S*  2x 

aze*  —  \  ea  <j>(a  —  2  x)  dx  -f  b. 
Replacing  a  by  its  value  x  +y,  we  have  the  general  solution 

2x  /•»  2x 

(x  -\-y)ze*+y  —  I  ga  <£(#  —  2  x)dx  =  $ 


236  DIFFERENTIAL  EQUATIONS  §86 

Here,  as  in  the  case  of  the  non-linear  partial  differential  equa 
tions  of  the  first  order  (§  81),  the  general  solution  cannot  be  written 
down.  For  until  the  form  of  <£  is  known  the  above  integral  cannot 
be  calculated.  In  any  example  in  which  the  initial  conditions  de 
termine  <f>,  the  a  which  appears  in  the  integral  must  be  replaced  by 
x  -\-y,  after  the  integration  has  been  effected. 


Ex.   4.     f 

Ex.  5.   ps  —  qr  =  o. 

Ex.  6.    (b  +  cqfr—  *(b  +  cq)(a  +  cp)s  +  (a-\-c£?t=  o. 


86.  Special  Method.  —  At  times,  by  considering  one  or  the  other 
of  the  independent  variables  as  a  constant  temporarily,  the  equation 
may  be  looked  upon  as  an  ordinary  differential  equation.  Of  course, 
an  arbitrary  function  of  the  variable  supposed  constant  must  take 
the  place  of  the  arbitrary  constant  in  the  solution.  The  following 
examples  will  illustrate  : 

Ex.  1.   xr=p. 

Letting  y  be  a  constant  temporarily,  this  may  be  written 

dp  dp      dx 

#-£=/,  or   -<£  =  —  • 

dx  p       x 

Integrating,  we  have  p  =  xf(y),  where  f(y)  is  an  arbitrary  function. 
Again  letting  y  be  constant,  we  have 


whence  a  =  x*f(y)  +  <fr(y),  where  <j>(y)  is  another  arbitrary  function. 
Here  the  factor  -,  arising  on  the  right,  is  incorporated  in/(y). 

2 


§86  PARTIAL,   HIGHER  ORDER  THAN  THE  FIRST  237 

Ex.  2.    r  +  s+p  =  Q. 
Integrating  this,  considering  y  as  a  constant,  we  have 


This  is  linear  and  of  the  first  order.      Hence  Lagrange's  method 

applies. 

dx  _  dy  _      dz 

i  "  i  ~~/00-* 
From  the  first  two  members  we  get 
x—  y  =  a. 

From  the  last  two  members  we  have  the  linear  ordinary  differential 
equation 

*+.=/«. 

An  integrating  factor  is  ey  (§  13),  and  the  solution  is 


Hence  the  general  solution  is 


or 

where  the  factor  e~y  is  incorporated  in  </>(j). 

Ex.  3.  yt—q  =  xf. 
Ex.  4.   s  =  xy. 
Ex.  5. 


238  DIFFERENTIAL   EQUATIONS  §87 

87.  General  Linear  Partial  Differential  Equations. — We  shall  con 
sider  now  partial  differential  equations  which  are  linear  in  the  de 
pendent  variable  and  all  of  its  derivatives.  The  general  type  of  such 
equations  is 

/T\        T>      dnz       „  dnz          p  dnz  p     dnz 

V1 )          •*«,  0  T~~:  ~T  -*il-l,  1  -|     n    1   -,       I    -«  n-2,  2  -     „    2  ~    g  T  *  "•  T  •*<),  n  T~~l 

5^*  &&l&y  dxn~2dy2  dyn 


Z    -L  P  S         Z  J_    P         9Z 

-i,  o  >  _  ,  H r  A  r  a  .  ,  r  H r-Ti,.«-r- 

"  r  dx 


where  tne  coefficients  are  functions  of  x  and  y,  including  the  case 
where  some  or  all  of  them  are  constants. 

If  we  put  D=  —  ,  ^  =  —  ,  (i)  may  be  written 
dx  ay 


or  more  briefly 
(i) 


where  ^(A  3)  is  a  symbolic  operator,  which,  looked  upon  algebra 
ically,  is  a  polynomial  of  degree  n  in  D  and  A  There  are  many 
points  of  similarity  between  this  equation  and  the  linear  ordinary 
differential  equation  of  the  nth  order  (§  42). 


Obviously,     F(D,  £)(u  +  v)=  F(D,  £)u  +  F(D,  3)  v. 


§  88  PARTIAL,   HIGHER  ORDER  THAN  THE  FIRST  239 

Hence  the  problem  of  solving  (i)  can  be  divided  into  two,  viz.  that 
of  finding  the  general  integral  of 

(2)  F(D,£)z=o, 

which  we  shall  call  the  complementary  function  of  (i),  and  that  of 
finding  any  particular  integral.  The  sum  of  these  will  give  the  gen 
eral  integral  of  (i). 

88.   Homogeneous  Linear  Equations  with  Constant  Coefficients.  — 

Following  a  generally  adopted  convention,  we  shall  use  the  term 
homogeneous  to  apply  to  an  equation  in  which  all  the  derivatives  are 
of  the  same  order.  In  this  case  the  symbolic  operator  is  homoge 
neous  in  D  and  «&.  Suppose,  besides,  that  the  coefficients  are  con 
stants,  and  the  right-hand  member  zero.  Our  equation  will  be  of 
the  form 

(1)  (k,D*  +  k^-lJb-  +  -  +  Vi^1-1  +  kjr)*  =  o, 

or 

Since  for  <£  any  function  whatever 

.ZX^O'  +  mx)  =  mr<j>(r+'\y  -f-  mx), 

where  &r+*(y  +  mx)  means  -f**'^  +  ***)  ,  the  result  of  substitut- 

\_d(y  +  mx)]r+> 

ing  z  =  $(y  +  mx)  in  (i)  will  be 

4>M(y  +  mx)J?(m,  i)=o. 
Hence  z  =  <f>(y  +  mx)  will  be  a  solution,  provided  F(m,  i)=  o;  /.<?. 

(2)  k&T  +  ^ai"-1  +  -  -  -  +  k^m  +  kn  =  o. 


240  DIFFERENTIAL  EQUATIONS  §89 

If  the  roots  of  (2),  which  we  shall  speak  of  as  the  auxiliary  equa 
tion,  are  distinct,  say  m^  m2,  •••,mM 


will  be  a  solution.     Since  it  contains  n  arbitrary  functions,  it  will  be 
the  general  solution.* 

Thus  let  us  consider  the  equation  in  Ex.  2,  §  85, 


_ 
~ 


The  auxiliary  equation  in  this  case  is 
m^  —  az  =  Q.     .'. 

Hence  the  general  solution  is 


dxdy 


v  z 

EX.2.      ——7 


dx*         dx?dy 

89.  Roots  of  Auxiliary  Equation  Repeated.  —  If  any  of  the  roots 
of  the  auxiliary  equation  are  repeated,  the  method  of  §  88  fails  to 
give  us  the  general  solution.  In  this  case  we  proceed  by  a  method 
entirely  analogous  to  that  in  §  47. 

*  If  F(D,  J$)  contains  ^  as  a  factor,  F(m,  i)  is  only  of  degree  «—  i.  The  lost 
root  in  this  case  is  oo  ,  and  the  corresponding  integral  is  <t>(x).  This  is  also  obvious 
from  the  form  of  the  differential  equation  in  this  case.  For  to  say  that  ^  is  a  factor  of 
f(D,  «$)  is  to  say  that  every  derivative  of  z  is  taken  at  least  once  with  respect  to  y. 
Hence  z  =  <£  (x)  will  give  o.  Similarly,  if  F(Dtj$}  contains  ^ras  a  factor, 
^2(^)1  ••*>  yr~l<t>r(x)  are  readily  seen  to  be  integrals. 


§  89  PARTIAL,   HIGHER  ORDER  THAN  THE  FIRST  241 

The  symbolic  operator  F(D,  3)  may  be  written  as  the  product  of 

its  factors 

(D  -  m^)(D  -  m*$)  .-(£>-  mj>). 


Moreover,  it  is  readily  seen  (§  46)  that  the  order  of  these  factors  is 
immaterial.  Suppose  m±  a  repeated  root.  We  wish  to  find  a  solu 
tion  of 

(D  -  mJ$)(D  -  mi$)z  =  o. 


Putting  (D  —  mleb)  z  =  v,  our  equation  is 

(D-m^)v  =  Q. 

Hence,  by  the  method  of  §  88  v  =  <j>(y  +  m^x). 

We  now  have  to  solve  (D  —  tnla$)z  = 

This  is  linear  and  of  the  first  order, 


Hence  the  method  of  Lagrange  (§  79)  applies, 

—  —  _  ty- 
i  mj_ 

From  the  first  two  members  we  have 

y  +  m\x 

Putting  this  in  the  last  member,  we  have 


Hence  the  general  solution  is 


or  z  — 


242  DIFFERENTIAL   EQUATIONS  §90 

In  other  words,  if  ml  is  a  double  root  of  the  auxiliary  equation,  not 
only  is  <$>(y  +  m\x)  an  integral,  but  so  also  is  xty{y  -\-m-iX).  In  an 
entirely  analogous  manner  it  can  be  shown  that  if  ml  is  an  r-fold  root, 


are  all  integrals. 


Ex.2.          + 

d.*3     ftndy     Sliwjjr 


Ex'3'       ~ 


90.  Roots  of  Auxiliary  Equation  Complex.  —  If  the  coefficients  in 
the  differential  equation  are  real,  the  complex  roots  of  the  auxiliary 
equation  occur  in  pairs  of  conjugates.  Hence  if  a  +  i(3  is  a  root, 
a  —  tft  will  also  be  one.  The  corresponding  terms  in  the  comple 
mentary  function  will  be 


<f>(y  +  ax  +  iftx)  +  $(y  +  ax-  iftx). 

<#>!  and  fa  being  any  two  arbitrarily  chosen  functions,  there  is  no  loss 
in  putting 


Our  expression  above  becomes  then 
fa(y  +  ax  +  ifix)  +  <h(y  +  ax-  iftx) 

+  ilfa(y  +  ax.+  iftx)  -  fa(y  +  ax-  / 
For  </»!  and  fa  any  real  functions,  this  is  real. 


§91  PARTIAL,  HIGHER  ORDER  THAN  THE   FIRST  243 


The  auxiliary  equation  is 

m2—  2m  +  2  =  0.     .-.m=i±z. 

The  general  solution  is 


It  will  assume  a  real  form 

*  =  <£  i  (y  +  x  +  ix)  +  <#>!  (  y  +  x  -  ix) 

+  /[^(>  +  *  +  /*)-  fcOy  +  *-**)], 
for  </>!  and  ^  any  real  functions. 

For  example,  if,  in  particular,  we  choose  <f>i(u)  to  be  cos  u,  and 
ty\(u)  to  be  eu,  we  have 

cos  (y  +  x  +  ix)  =  cos  (jc  +  y)  cos  /jc  —  sin  (x  +  j)  sin  ix 

=  cos  (jc  +  y)  cosh  #  —  /  sin  (A:  +  y)  sinh  ^, 

cos  (  y  +  ^  —  MC)  =  cos  (x  +  jv)  cos  /a:  +  sin  (jc  +  y)  sin  «c 
=  cos  (x  +  7)  cosh  #  +  /  sin  (^  +j^)  sinh  x. 


.'.2  =  2  cos  (x  -\-y)  cosh  ^  —  2  ^r+v  sin  x. 

91.  Particular  Integral.  —  General  methods  for  finding  the  par- 
ticular  integral  which  must  be  added  to  the  complementary  function 
to  get  the  general  integral,  in  case  the  right-hand  member  of  the 
equation  is  different  from  zero,  may  be  deduced  along  lines  entirely 


244  DIFFERENTIAL   EQUATIONS  §91 

analogous  to  those  for  linear  ordinary  differential  equations  with  con 
stant  coefficients  (§§  47,  48).*  In  a  large  number  of  cases,  these 
can  be  found  more  simply  by  trial,  by  methods  similar  to  that  of  un 
determined  coefficients  (§  50).  The  following  examples  will  illustrate  : 

,32  ,32  .32 

Ex.  1.    -—+  —  ^  --  2  -^  =  sin  (x  -f  2y)  —  2  sin  (x  -f-  y)  +  x  -f  xy. 


The  complementary  function  is  $(y  +  x)  +  \l/(y  —  2  x). 
To  get  sin  (x-{-  2  y),  since  all  the  derivatives  are  of  the  second 
order,  we  try  z  ,=  a  sin  (x  -f  2  y). 

Then  |5  j  +   *L  -  ,  «*  :  =  sa  sin(*  +  ,y). 
dx2      dx  dy         dy2 

If  a  =  |,  this  becomes  sin  (x  +  2  y).  Hence  the  required  particu 
lar  integral  is  i  sin  (x  +  2  y). 

Since  sin  (^+j)  is  part  of  the  complementary  function,  there  is 
no  use  in  trying  z  =  ^sin  (x-\-y).  Trying  z  —  bx  sin  (x-\-y)  we  get 
3/£  cos  (x  -\-  y).  Hence  we  must  try  z  =  t>xcos(x  -\-y).  Doing  this, 
we  have  as  a  result  of  substituting  in  the  equation  —  3&sin(x+y). 
This  will  be  —  2  sin  (x  +y)  if  b  =  f  .  Hence  -|  x  cos  (x  +y)  is  the 
required  particular  integral.  [It  is  obvious  that  we  might  have  also 
used  z=.by  cos  (x  -f-jOO  To  get  #,  we  try  z  =  r^3.  Substituting  this, 
we  get  6  ex.  This  equals  .#  if  6  <:  =  i  ;  hence  the  corresponding 

^3 

particular  integral  is  -  .     To  get  xy,  we  try  z  =fy?y.      Using  this  we 
6 


get  6fxy  +  $fx*.  So  we  try  fs  =s/&f*y  -h^aA  Using  this,  we  get 
6Ay  +  (3/+  i2^X.  This  equals  xy  if/=-J,  ^=—  ^  Hence 
the  required  particular  integral  is  J  ^  —  ^¥  jc4.  And  the  general 
solution  is 


*  Thus,  for  example,  see  Forsyth,  §  250,  Johnson,  §  320  and  foil.,  Murray,  §  130. 


§9i  PARTIAL,  HIGHER    ORDER  THAN   THE  FIRST  245 

Ex-2-      - 


The  complementary  function  is  <j>(y  +  x)  +  $(y+  2  x). 

To  get  ex+2y  we  try  z  =  aex+2y.     Using  this,  we  get  3  aex+2v. 

Hence  %  e*+2y  is  the  particular  integral  desired. 

Since  ^+y  is  a  part  of  the  complementary  function,  let  us  try  z  — 
Using  this,  we  get  —  be***.  Hence  —  xe"+*  is  the  particular 
integral  desired.  [Of  course,  we  might  have  used  z  =  t>ye*+y  instead.] 
And  the  general  solution  is 


Fx  3  Z 

dx2dy 

The  auxiliary  equation  is  m2  —  2  m  -f-  1  =  o.     .*.  m  =  i,  i. 

[We  have  here  an  example  of  the  case  cited  in  the  footnote  of 

88.] 

The  complementary  function  is  <j>(x)  +  $(y-\-x)  -\-x\(y-\-x). 

dsz  i 

Since  there  is  no  term  in  —  ,  in  order  to  get  —  ,  we  have  to  take 


y  times  a  function  of  x  which  on  being  differentiated  twice  gives  —z  > 

x 

that  is,  we  shall  try  z  =  ay  log  x.     Doing  this,  we  find  that  if  a  =  —  i 
we  get  i. 

Hence  the  general  solution  is 

z  =  <f>(x)  +  *t>(y  +  x)  +*\(y  +  x)  -y  log*. 


246  DIFFERENTIAL  EQUATIONS  §92 

92.  Non-Homogeneous  Linear  Equations  with  Constant  Coefficients. 
—  If  F  (D,  J$)  of  §  8  7  is  not  homogeneous  in  D  and  ^,  but  the  co 
efficients  are  constants,  it  is  only  under  certain  circumstances  that  we 
can  obtain  solutions  involving  arbitrary  functions,  although  we  can 
always  find  solutions  with  an  indefinite  number  of  arbitrary  constants. 

Since  jyeax+*y  =  areax+b\  &°eax+ly  =  b*eax^\  the  result  of  substitut 
ing  z  =  ceax+by  in  the  left-hand  member  of 


(i) 

is  cS**+lvF(a,  £).     If  a  and  b  satisfy  the  relation 

(2)  <F(M)=o, 

which  we  shall  call  the  auxiliary  equation,  z  =  ceax+ly  will  be  a  solu 
tion  of  (i)  where  c  is  any  constant.  Corresponding  to  any  value  of 
b  there  will  be  a  definite  number  of  values  of  a  satisfying  (2).  Hence 
we  can  find  as  many  particular  solutions  as  we  please  by  giving 
various  values  to  b.  Now,  the  sum  of  any  number  of  integrals  of 
(i)  is  also  an  integral.  Hence 

(3)  z  =  2cS*+*» 

is  a  solution  where  the  <r's  and  £'s  are  arbitrary  constants,  indefinite 
in  number,  and  each  a  is  so  chosen  that  with  the  corresponding  b  it 
satisfies  (2). 

If  corresponding  to  any  value  of  b  we  have  the  k  values  of  a  satis 
fying  (2)  in  the  form  f\(6)  ,/*(&),  '~,fk(fy>  we  can  write  (3)  m  tne 
form 

(4)  z  = 


where  the  r's  and  the  £'s  are  perfectly  arbitrary. 

In   general,   F(D,  &)    has   no   rational    factors.      If  there   is   a 
linear  factor  D  —  X^  —  p,  the  equation  (2)  will  contain  the  factor 


§92  PARTIAL,  HIGHER   ORDER    THAN  THE   FIRST  247 


a  —  \b  —  fi,,  whence  a  =  \b  +  /x.      Hence  one  of  the  /'s,  say 
becomes  \b  -f  /A,  and  the  corresponding  set  of  terms  in  (4)  may  be 
written 


Since  the  <r's  and  the  £'s  are  arbitrary,  2<r/(Az+J/)  is  an  arbitrary  func 
tion  of  Xx+y,  say  <j>(\x+y).  So  that  (5)  may  be  written 

(6)  z  =  #»4>(\x+y). 

Hence  we  see  that  corresponding  to  every  distinct  linear  factor  of 
(2)  we  have  a  solution  of  the  form  (6).* 

If  there  is  a  linear  factor  of  F(D,  3)  which  is  free  of  Dy  the  cor 
responding  factor  in  (2)  will  be  free  of  a  •  let  it  be  b  —  p.  The 
corresponding  set  of  terms  in  (4)  may  be  written 

(5  ')  z  =  *$ceax+w  =  e^ce™. 

Since  the  <r's  and  #'s  are  arbitrary,  ^ceax  is  an  arbitrary  function  of 
x,  and  (5')  may  be  written 
(6')  z  =  e»y<t>(x). 

If  the  right-hand  member  of  the  differential  equation  is  not  zero,  a 
particular  integral  may  frequently  be  gotten  by  trial  as  in  §  91. 

-r-,     .       82z  ,     82z     ,  dz  .    /  x 

Ex.1.    .—  +__+__  z  =  sin  (x+2y). 
dx*      dx  dy      dy 

The  auxiliary  equation  (2)  is 

a2  +  ab  +  b  —  i  =  o, 
or  (0+  i)(a  +  b  —  i)  =  o. 

*  In  the  case  of  the  homogeneous  equations  ($  88)  all  the  factors  are  linear.  Be 
sides,  in  that  case,  M  =  o,  and  the  result  there  obtained  coincides  with  what  we  have 
found  here. 


248  DIFFERENTIAL   EQUATIONS  §92 

Using  a+  i  =  o,  we  have  X  =  o,  //,  =  —  i,  and  from  (6)  we  see 
that  z  =  t~x<f>(y)  is  a  solution. 

Using    a  +  b—  i  =  o,    we    have    \  =  —  I,/A=  i.       Hence    z  — 
—  •*)  is  a  solution.      The  complementary  function  is,  therefore, 


For  a  particular  integral  try  z  =  a  sin  (x  +  2  j)  +  ft  cos  (^  + 

Substituting  this  in  the  left-hand  member,  we  get 

(—  4  «  —  2  /?)  sin  (x  +  2^)  +  (—4  /?  H-  2  a)  cos  (x  +  2  j). 


This  will  equal  sin  (x  +  2  y)  if  «  =  —  -  ,  ft  =  —  —  .     Hence  a  particu 
lar  integral  is 

—  -  sin  (x+zy)-—  cos(>+2>>), 
5  I0 

and  the  general  solution  is 

z  =  e~x<f>(y)  +  ^(y  -  x)  -  —  [2  sin  (#  +  2  y)  +  cos  (^  + 


E,2.         _8 


The  auxiliary  equation  (2)  is 


or 


Hence  the  complementary  function  is 


§93  PARTIAL,  HIGHER  ORDER  THAN  THE   FIRST  249 

Since  e~*  is  part  of  the  complementary  function,  we  would  naturally 
try  z  =  axe~x.  But  this  is  also  part  of  the  complementary  function. 
as  may  be  seen  by  putting  <j>  = 


~  X  ,  ty  =r- 


We  must  then  try  z  =  a#V~*.  Substituting  this  in  the  left-hand 
member,  we  get  2  ae~x.  Hence  a  =  -  ,  and  the  particular  integral 
desired  is  -x2e~x.  The  general  solution  is 


z 

Ex.3.    ^- 

dx2 

Ex.4.       ?_ 


)  +  ± 


dy      dy 


93.  Equations  Reducible  to  Linear  Equations  with  Constant  Co 
efficients.  —  If  the  coefficient  of  £>r£s  in  F  (D,  £)  of  §  87  is  a  con 
stant  times  xry",  the  equation  can  be  reduced  to  one  with  constant 
coefficients  by  the  transformation  log  x  =  X,  log  y  =  Y.  (Compare 
with  Cauchy's  equation,  §  51.)  Thus 

Dz-dz  -1  dz  '  xDz-  dz 

~~'  ~dX' 

dz  .        /2        322        dz 

'dX2~'dX' 


dz      i  dz  ,         dz 

—  =  ~Tl>i  ••^3*as-TT> 

dy     y  dY  dY 


--  --- 

y2dY2     y2dY'  '  dY2     dY' 


xydXdY'  '  '  dXdY 


25O  DIFFERENTIAL   EQUATIONS  §93 

Ex.l.     ^ 


Making  the  substitution  log  x  =  X,  log  y  —  V,  the  equation  becomes 

d2z    ,         d-z       ,    d2z        dz        dz        9r  ,     9V. 

'  whlchhas  constant  c°- 


efficients.     The  auxiliary  equation,  (2),  §  92,  is 

a?  +  2  ab  +  P  —  a  —  b  —  o,  or  (a  +  b}(a  +  b  —  i)  =  o. 

Hence  the  complementary  function  is  <f>(Y—X)  +  ex\\i(Y—  X). 
For  the  particular  integral  try  z  =  aezs  +  /3^r.    Doing  this,  we  find 

that  a  =  (3  =  -.     Hence  the  general  solution  is 

2 


Passing  back  now  to  x  and  y,  and  remembering  that  Y—  X=  log--, 
the  general  solution  takes  the  form 


s»  — 


[This  equation,  being  linear  in  r,  s,  t,  comes  under  the  head  of  the 
case  treated  in  §  85.  The  student  should  solve  this  example  by 
Monge's  method,  as  an  exercise.] 


Ex.2.    *^y +»•-., 

dx2          d  dx         d 


Ex.3.     ^- 

dy 


§94  PARTIAL,  HIGHER   ORDER  THAN  THE  FIRST  251 

Other  equations  may  be  reducible  to  linear  equations  with  constant 
coefficients.  (But  the  transformation  is  not  always  so  obvious  as  in 
the  case  cited  above.)  Thus  let  the  student  apply  the  transformation 
X  =  -^,  K=-/  to  the  following  example. 

2  2 


F       4 


94.  Summary.  —  The  number  of  classes  of  partial  differential 
equations  of  higher  order  than  the  first  which  can  be  integrated  by 
elementary  means  is  very  small.  In  this  chapter  we  have  dealt 
almost  entirely  with  differential  equations  either  linear  in  the  depend 
ent  variable  and  all  of  its  derivatives,  or  linear  in  the  highest  deriva 
tives  only,  these  being  of  order  two.  This  latter  class  frequently 
yields  to  Monge's  method,  §  85. 

If  the  equation  is  linear  in  the  dependent  variable  and  all  of  its 
derivatives,  and  has  constant  coefficients,  the  general  method  of 
§  92  applies. 

If  the  linear  equation  with  constant  coefficients  is  "  homogeneous," 
that  is,  if  the  dependent  variable  is  absent,  and  all  the  derivatives 
that  appear  are  of  the  same  order,  the  method  of  §  88  applies. 

If  the  equation  is  linear,  but  the  coefficients  are  not  constants,  a 
transformation  can  sometimes  be  found  to  reduce  the  equation  to 
one  with  constant  coefficients  (§  93). 

At  times  the  special  method  of  §  86  can  be  applied  directly  to  an 
equation. 

Ex.  1.  ys  =  x+y. 

Ex.  2.  r  —  s—  6  t=xy. 

Ex.  3. 

Ex.  4.  xr- 


252  DIFFERENTIAL   EQUATIONS  §94 

Ex.  5.   xr—p  —  xy. 

Ex.  6.    r-  t-  3/  +  3  ?  =  **+*. 

Ex.  7.   *V-//=(#+i)j>. 

Ex.  8.   #V  +  2  xys+ft+xp+yq  —  2  =  0. 

Ex.  9. 


Ex.  10.   j-/=-. 


NOTE  I     CONDITION  FOR  RELATION  BETWEEN  FUNCTIONS       253 


NOTE   I 

Condition  that  a  Relation  exist  between  Two  Functions  of  Two  Vari 
ables.  —  If  u  and  v  are  two  functions  of  x  and  y,  the  necessary  and  sufficient 
condition  that  a  relation  exist  between  them  is  that  their  functional  determi 
nant  (also  called  their  Jacobian)  vanishes,  that  is, 


du  dv 
dx  d* 
du  dv 

dy  dy 


=  0. 


=A*(*,«0  = 


1°  To  prove  the  necessity  of  the  condition. 
If  u  =  0(^),  differentiating,  we  get 


dx~  dv 


dy     dv  dy 

These  two  equations  in  the  single  quantity  —  can  hold  simultaneously  only  if 

dv 


du  dv 

dy  dy 


=  o, 


which  proves  the  necessity  of  the  condition. 

2°  Now  to  prove  the  sufficiency  of  the  condition. 
Suppose  u  and  v  to  be  given  by 


From  these  we  can  eliminate  y,  resulting  in  a  relation,  which  may  be  supposed 
solved  for  «,  thus 

u  -  <t>(x,  v). 


254  DIFFERENTIAL   EQUATIONS  NOTE  II 

Differentiating,  and  remembering  that  x  and  y  are  independent  variables,  we 
have 

dx     dx  dv  dx 

du (90  dv_ 

dy  dv  dy 
du  dv 

(\X    r\X 

If  .      =0,   these    equations    can    hold    simultaneously   only   provided 

dy  dy 

&£  =  o.     But   this  means  that   0  is   free  of  x.     Hence,   when  the   Jacobian 

vanishes  u  =  0(#), 

which  proves  the  sufficiency  of  the  condition. 

Remark.  —  This  theorem  can  be  extended  to  n  functions  of  n  independent 
variables. 

NOTE   II 

General  Summary  ^ —  The  following  is  an  index  to  the  various  methods, 
given  in  this  book,  for  solving  differential  equations : 

In  the  case  of  a  single  ordinary  differential  equation, 

if  it  is  of  first  order  and  first  degree,  see  §  19; 

if  it  is  of  first  order  and  higher  degree  than  the  first,  see  §  28  for  the  general 
solution,  and  §  34  for  the  singular  solution; 

if  it  is  of  higher  order  than  the  first  and  linear  with  constant  coefficients,  see 
§  52  (note  what  is  said  there  of  a  very  general  class  of  linear  equations  which 
can  be  transformed  to  linear  equations  with  constant  coefficients) ; 

if  it  is  of  the  second  order  and  linear,  see  §§  55,  62,  and  74; 

if  it  is  of  higher  order  than  the  first  and  does  not  come  under  any  of  the  above 
heads,  see  §  62. 

If  there  is  a  system  of  ordinary  differential  equations,  see  §  69. 

As  a  final  resort,  whether  there  is  a  single  ordinary  differential  equation  or  a 
system  of  them,  the  general  methods  of  Chapter  XI  may  be  tried. 

If  there  is  a  single  total  differential  equation  in  more  than  two  variables, 
see  §  41. 

If  there  is  a  single  partial  differential  equation  of  the  first  order,  see  §  84. 

If  there  is  a  single  partial  differential  equation  of  higher  order  than  the  first, 
see  §  94. 

The  above  may  be  put  in  the  following  tabular  form : 


NOTE  II 


GENERAL  SUMMARY 


255 


•r* 

1     « 

4     i 

I 

& 

Tj- 

cr1 

1 

00 

o> 

£ 

-4 

i 

i 

O 

iS^ 

CO 

.                | 

oT                       c^» 

1 

1 

?           2 

^            1 

1 

1 
W 

1           | 

'C              +•» 

>                rt 

C^>                                          -M 

*s  s              ^: 

TD  ^3                                   M 

:erential 

3 

§ 

^N 

—  J^                                   ^ 

§ 

1~ 

G 

3 

'S 

0 

So                       '-g 

.S     U                                          <U 

& 

"^    w                                    oJ 

5 

S 

o 

I 

IS 

Q 

o 

a 

d 

S 
.S 

IS 

la           1" 

f 

80 

"73 

g**                         Tj-                  .2 

o 

"o 

o  c/r          ^0       ?p 
L5S        ^o 

^1           1          o 

OH 

rt 

So            £ 

>—  i 

X 

1* 
•e 

1    « 

g       -1 

1 

U 

S 

<*-, 
o 

»-.                 oo 

'o 

<u                  o 

I-) 

'bjO                  ^^ 

OJ 

}_l 

0 

IS                tf 

S 

"S 

.2 

_, 

_o  

"3 

2 

2 

o\                o 

G 

IB 

C«7J                             *c3 

1 

ANSWERS 


Section  3 


dx 


Section  8 


4. 


Section  9 

2.  y  +  2  xy  —  2  x^  =  cx*y. 

4.  sec*  +  tanjj>  =  r. 

Section  10 

3.  log  x  —  —  =  c. 

4.  y*  =  cx^  (x2  -\-  y2-). 

5.  *2  +  y2-  =  cxtyt. 

6.  log  x  —  sin  ^  =  r. 

x 

Section  11 


3.  5*- 


Section  12 


y     xy 

Section  13 


3.  2y=(x  + 

4.  y(i  +x*V 


I)2. 


5.       = 


2.  (jf/2- 

3. 

4. 

5. 


Section  14 


I  ss  *  +•  i  4.  ^V*  +  i. 

Section  15 

2.  yi  =  ex. 

3.  *3  +  y  =  «ry2. 

Section  16 

4.  *a-y=,.r. 


Section  17 


257 


258 


ANSWERS 


y  = 


Section  18 


if  £  and  c  have  the  same  sign; 


if  b  and  <:  have  opposite  signs. 
Section  19 


23.  ^r  +  /  -  4  log  (2  ^  +  $y  +  7) 

24.  *y  (^2  -  jr2)  =  c. 
25. 


26.    Vi 
27. 


=  c. 

+   I   +  CX. 

y 

3.  x^y2-  —  2.  xy  log  cy  =  I. 

.* 

Section  20 


ANSWERS 


259 


Section  21 

never  vanishes  at  a  finite 

time,  while 

6.  The  parabolas  y2-  =  2  X*  +  c. 

the  distance  covered  increases  con 

i 

tinually,  but  never  attains  the  value 

7.  y  =  ce~x. 

°at  a  finite  time. 

8.  The  circles  x1  +  y2  =  ex. 

k 

^ 

9.  The  cardioids  p  =  c(i  —  cos  0). 

-—t                p 

h  ce  ~J?, 

10.  The  spirals  />2  =  ce±e. 

R. 

Rt 

Section  22 

E 

6.  The   circles  x^  +  jj/2  =  c  (my  —  x}, 

(see    Ex.  6,  §   17),  through  the  ori 

Section  24 

gin  with    their  centers  on  the   line 

4.  y2  =  2  x  (x  -  o2. 

y  =  —  #/.*•,  where  m  =  tan  ct. 

7.  The    logarithmic    spirals    pe1**  =  c 

5.  2.x  =  ce»  —• 

(compare  Ex.  5). 
8.  The  ellipses  2x*+y*  =  e. 

6.  (i+xy-cy)(x  +  y+i-ce*') 
(x  —  y  —  i  —  eg-*}  =  o. 

9.  The  equilateral  hyperbolas  xy  =  c. 

10.  y*  =  cxb. 

Section  25 

12.  p  =  c(i  +  cos  6}. 

4     --r      £  +  <?X 

13.  p=/c~r]p. 

C  -  6** 

14.  pTO  cos  w#  =  £•"*. 

5.  xy  —  fix  +  c. 

15.  tan  0  —  ce*t>. 

6.  4(*2  +  y)3  =  (2  x3  -f  3  A 

y  +  O2. 

1C                     2t            tl_    r  .,..    _r 

Section  26 

I  -f  COS  6 

olas,  confocal  and  coaxial  with  the 

1  y            c          x     ~cp 

^2/2  +  3) 

original  family  (compare  Ex.  4). 

0+/2)*               (i 
2    v2      2  ex  -\-  #2^2      o 

+  /2)1 

X  cos  6  —  c 

4.  jy  =  f  (.*  —  <r)2. 

conies  (compare  Ex.  3). 

5.  The  family  of  circles, 

Section  23 

\x  —  c) 

2  +y  =  « 

2.  z/=£?sina;  x  —  XQ  —  -  gt*  sin  a. 

Section  27 

2 

4.  ^  =  c(?  +  i)  +  A 

o     ~.       „    g  lorr^r*  ~l~^~rt 

5.  y*  =  6*2  +  -  . 

0                         Q              O                           . 

/"^                         C    "|~    I 

e 

5.  x  —  XQ  =  ~  (ekt—  i).    If/^>o,  the 

A 

velocity  and  the  distance  covered  in 

4 

79                               L       •     ~fl 

crease    indefinitely  ;     if  k  <  o,  the 

.  y^  :=  ex  -\  —  r*. 
8 

velocity  diminishes  continually,  but 

9.  y  =  ex*  +  fix. 

260 


ANSWERS 


Section  28 

1.  (x  -  O2  +  f  =  a*. 

2.  cy  =  c\x  -  b)  +a. 

3.  x  +  cxy  +  <*  =  o. 

4.  x*+-c(x-$y)  +  c2 

5.  -/ 


7.  y(i  ±  cos*)  =  c. 
g.  {y  _  CXY  =  i-c2. 

9.  0/-«:)2  =  4  A 

10.  .y2  =  ex2  +  A 

11.  x+2cye*  =  c2xe2x. 

12.  0  = 


y  —  —  —  ;   or  putting/  =  tan-, 


=  -(i  -cos  0), 

2 


*=* 

2 

a  family  of  cycloids  generated  by  a 
circle  of  radius  -•     We  see,  then, 

2 

that  a  characteristic  property  6f  a 
cycloid  generated  by  a  circle  of 
radius  a  is  that  s2  =  8  ay,  where  s 
is  the  length  of  arc  measured  from 
the  nearest  cusp. 
15.  The  same  family  of  cycloids  as  in 


Ex.  14. 


1.  y2  = 


Here  k  =  -  g< 
Section  29 


Section  32 

y>_ 


— 1- -21=  i,     when    the     fixed 

C^   ~]~  £  K 

points  are  (<r,  o),  (—  c,  o),  and  £  is 
the  constant  product.  According 
as  k  is  positive  or  negative,  we  have 


the  ellipse  or  the  hyperbola,  having 
the  fixed  points  for  foci,  and  ^J  ±k 
for  semi-conjugate  axis. 

2.  The  circle  x'2  +  y2  =  &*,  where  k  is 
the  constant  distance. 

3.  The  equilateral  hyperbola  2xy=.  a2. 

4.  The  parabola  (x  —y}2  —  2  a(x 


5. 


+  4<-  =  o,  g.s. 
xy  —  1  =  0,  s.s. 


(y  +  x-  c}2=4xy,  g.s. 
ory  =  O,  s.  s. 


Section  33 


4.     .r- 


=^->  g-s-»  ^  =  i»  s-s-i 
^  =  o,  c.  1.,  y  =  \,t.  1. 
§24,  Ex.  3,j2-  i  =o,  s.s. 

Ex.  4,  there  is  no  s.  s.,  .*  =  o, 
t.l.  and  p.s.  for  <r=  00,^=0,  n.l. 
Ex.  5,  jr2  4-  i  =  o,  s.  s. 
§  25,  Ex.  5,  4*V+  i  =o,  s.s., 

x  =  o,  1.  1. 
Ex.  6,  there  is  no  s.  s. 

x*+y  =  o,  c.  1. 
§  26,  Ex.  2,  x2  -  a2y2  =  o,  s.  s. 
Ex.  4,  4  *3  -  277  =  o,  s.  s., 

y  =  o,  s.s.  and  t.l. 
§  27,  Ex.  2,  *2  +  4  *2y  =  o,  s.  s. 
Ex.  6,  *y  =  o,  s.  s., 


§  28,  Ex.  2,  y2  -  4  a(*  -  £)=  o,  s.& 

Ex.  3,  xy*  -4  =  0,  s.  s., 
.*•  =  o,  p.  s.  for  c  =  o. 

*  In  these  answers  the  following  abbre 
viations  are  used  :  g.  s.  for  general  solu 
tion,  s.s.  for  singular  solution,  p.s.  for 
particular  solution,  1.  1.  for  tac-locus,  n.  1. 
for  nodal  locus,  c.  1.  for  cuspidal  locus. 


ANSWERS 


26l 


Ex.  4,  x2  4-  2  xy  -  3y=o,  s.s. 

Section  45 

i          Ex.  5,  (x  4-  i  )j  =  o,  s.  s. 

2. 

y  =  (c\  +  ^0  cos  x  +  O  +  <-4.*)  sin  ^ 

Ex.  11,  A:2  +y!  =  o,  s.s., 

. 

f           ,  =  g,M. 

3. 

v  =  ri  4-  ^^x  (  C2  cos  £  -v/3 
\           2 

5.\4^2  =  4*  4-  i,  s.  s.,    ^  =  o,  1.  1. 

i 

2          J 

Section  36 

x 

Section  47 

5*  *  =  «*• 

2. 

y  -  cie~x  4-  ^2^"2z  +  <?-2z<?'Z- 

1  ^2+y  4-(s-<:)2  =  a2. 
4.  *2  4-  jr2  4-  z*  =  f*. 

3. 

/                              0    .    20^-^ 

_y  •_!!    1    t'j^  4"  i'2-^'  4"  ^'3-t       4"                  r-       ""             1  ^      " 

> 

CJ^^^A.---^-      O  T 

2a!   ,  cos  JT  —  3  sin  ^r 

oec  tion  o  / 

. 

y  —  ^i^      T  ^2^      i  

10 

1.  xy  +  yz  +  zx  =  c(x  +  y  +  z). 

5. 

y  =  (ci  4-  c<ix}  e*  —  e*  log  (i  —  *). 

^y_±z  +  z±x  =  ^ 

Section  48 

Section  41 

1. 
2. 

^  =  CT,f  4-  ^2^2Z  ~  XP. 

1.  yz  4-  zx  4-  xy  =  c. 

9  A:2  4-  6  x  4-  20 

2.    *2  4.  j2  _  ,  (z  _|_  !)2. 

27 

3.  ^(x  +y2  +  22)  =  ^. 

3. 

_y  =  <TI  cos  jr  4-  cz  sin  ^  +  ^  sin  x 

41-                      Z             _L    y 

4-  cos  .*•  log  cos  x. 

.      #  -f-    t. 

4. 

y  =  (*TI  4-  c<>x\  e*  -f-  £"3^2z  •  < 

5.   f(y  +  z)  =  c. 

2      4 

6.    (*2  4-  j/  +«)***  =  *-. 

Section  49 

7.    X*  +  AT/  +  ^T22  -  /  =  C. 

2. 

y  =  c\  cos  x  4  ^2  sin  x 

_v 

8~ 

—  cos  .#  log  (sec  x  4-  tan  #). 

.  z  —  ce  x. 

9.    (x  —  i  )(jy  —  i)  (z  —  i)  **+»+*=£. 

10.   (2_^)2(^  +  2jJ/)-r. 

Section  50 

11.   (^  4-  z)  e*+y  =  ct. 

3. 

y  =  <T!  cos  .#  +  ^2  sin  x  4-  ** 

12.   (7  4-  2)  (/  4-  <:)  -f  2  (^  —  /)  =  0, 

+  x3  -  7  *. 

Section  43 

4. 

_y  =  (c\  4-  rgJf)^"36  +  -^2z  —  -sin  JT. 

3.  y  =  c\  4-  ^2^"  +  <^~z' 

5. 

^  =  ^^  +  H«(fl,C08-V3 

4.  y  —  c\e*  -|-  cze~x  +  ^3<?2x. 

%     -\ 

Section  44 

4-r3sm-V3j-^ 

6. 

i 

2.  _y  =^  ^*  (fj  4~  ^'2'^'/  4"  ^3^~z- 

3 

3.  y  =  ^~z(^i  4  ^-^  4-  Oj-*"2)  4~  ^4^*. 

2              14             I                      I 

H  —  jr2  -x-\  sin^r+     cos. 

4.  y  —  c  i  4  e*x  (r2  4-  ^3-^)' 

3           9          10              5 

262                                                    ANS1 

9.  y  =  (ci  4-  c^x}  cos  x 

4-  (^3  4  c\x}  sin  x  —  -  x2  cos  x. 
8 

Section  51 

2.  y  =  (^i  cos  log  .r  4-  <r2  sin  log  x}  x 

kVERS 

9.  ^  =  fX  sin  2  .*•  4-  r2  cos  2x 
_L  I  —  :rsin  2:r 

8 

10.  _y  =  a  cos  (*  4-  ^)  4- 

sin  JT  log  (tan  *  4-  sec  *)  -  i. 
11.  ^  =  (£l  4.  r2jr  +  fg^2)  <#  _  x  _3 

120 

12.  _y  =  (<TX  4-  cix  4-  ^s^r2)  <?z  +  r^"2* 

40 
13.  y=-c\  cos  x  +  c%  sin  .*• 

jr2  sin  jr  4-  or  cos  jr 

_a^logV     i*         x 

X                     X       &  I  —  X 

4 
jr        4  jr 

--  /                X 

15.   v  =  r  i^*  4-  ^"2  1  ^2  cos  -  v^ 

\              2 

2/6 

cos  2  JT  4-  8  sin  2  ^      i 

"*        6 

Section  52 
1    y—fl,2x<c^x       cos  AT  —  sin  x 

10 

-2-  y  =  fi**  -f  f2*?~z  4-  ^3  sin  x  -\-  c±  cos  .*• 
<?*  cos  jr. 
$.  y=  (<T!  4-  <:2;ir)  <r~z  +  2  jr3  —  12  *2 

_|_  ->g  -*•         AQ         V     **         lif 

130                       2 

16.  y  —  (c\  4-  f2jr)  f*  sin  jr 

2 

18.    i°.  £  >  *.      0  =  r*t(AeP*  4-  ^-M'> 
where  /x2  =  £2  —  »2.     6  diminishes 
continually  but  vanishes   only   for 
t  =  oo  ,  theoretically.      Practically, 
the  swing  of  the  pendulum  is  soon 
damped  sufficiently,  so  that  the  as 
sumed  law  ceases  to  hold,  and  d  be 
comes  zero  in  a  finite   time.     But 
the  solution  tells  us  that  the  pen 
dulum  comes  to  rest  without  vibrat 
ing. 
2°.    k<n.  0=Ae-ktcos(fj.t+J5), 
where   /x2  =  n2  -  k\     The  pendu 
lum  vibrates  with  constant  period 

27T 

24 
12        8 

-  g.r*2*. 

6.  y  —  (c\  4-  *>.*)  e?  4-  (^3  4-  f4Jr)^~x 

4-  -  cos  x. 
4 
t-  y=  (fi  4-  ^2  log  *)  sin  log  jf  -{- 
(ca  4-  ^4  log  JT)  cos  log  x  +  (log  jr)2 
4-2log*-3. 

3        2 
+  8*. 

/—  2  7^  >  which   is  longer    than 

ANSWERS 


263 


in  the  case  of  motion  in  a  vacuum 
(Ex.  17).  The  amplitude  is  Ae~Mt 
which  diminishes  continually. 

3°.  k-n.  6  =  (A  +  Bf)  e~*. 
If  A  and  B  have  the  same  sign, 
6  diminishes  continually,  as  in  case 
i°.  If  A  and  B  have  opposite 
signs.  6  passes  through  zero, 
changes  sign,  attains  a  maximum 
in  absolute  value  and  then  dimin 
ishes  continually  in  absolute  value, 
as  before. 


r 


cos  mt. 


2°.     m  —  n.      6  =  Acos(nt+  B} 

+  —  sin  nt. 
2n 

(<$)  The  same  complementary  func 
tion  as  in  Ex.  18* 


This  last  term  may  also  be  written 
cos  (mt—a), 

where  tan  a  =  — — — .     The  part 
n2  —  m2 

of    the    motion   indicated    by   this 
term  is  called  the  forced  vibration. 


Its  period   is  —. 


It   is    not    in 


phase  with  the  periodic  force  [ex 
cept  in  case  (a)  i°],  lagging  be 
hind  by  the  angle  a.  When 


o<a 


-<«<7r 


<-; 


when 
when    n  =  m, 

In    this    case   the    forced 


«  =  -. 
2 
vibration  is  given  by  U  —  —  sin  nt 

2k 

for  case  (£),  and  by  U=  —  sin  nt 
2  n 


tor  case  (a).  If  k  is  small,  the 
amplitude  is  large  in  case  (£),  while 
in  case  (a)  the  amplitude  increases 
with  the  time.  This  explains  reso 
nance  in  Acoustics,  the  effect  of 
the  measured  step  of  soldiers  on  a 
bridge,  and  the  like. 

20.  x  -  XQ  cos  kt+  —  sin  kt.    The  period 
is  independent  of  XQ  and  VQ. 

21.  2  x= 


If  c\  =£-.  o,  the  first  term  soon  predomi 
nates,  and  the  motion  is  spiral. 

If  c\  =  o,  the  second  term  soon 
becoming  negligible,  simple  harmonic 
motion  results. 

When  ro  =  o  and  V   =  -&- 


r  =  -£—sin 
2  w2 


=  o  and 

2u 
u>/;     i.e.    we   have    simple 


harmonic  motion  from  the  beginning. 

Section  53 

2.  y  =  c\P  +  czxte*  -f  x. 

3.  y  —  c\x  +  t2(i  +  .rtan-1.*:). 

4.  y  —  cix  +  c^e*  +  x*  +  I. 
6.  y  =  sec  x(cieax  +  c^e-**). 


i  _x! 
7.  y  =  x  e    4 


+  <r2  log 


8.  ^ry  =  ^^  +  c^e 

Section  54 

2.  j  =  ^i^  Vi  —  .r2  +  <r2(i  —  2  *2). 

3.  /  =  fi  sin  (sin  ^r)  -f  c2  cos  (sin  ^r) 


*  Replace  k  by  km  throughout  the  answer. 


264 


ANSWERS 


Section  55 

Section  61 

J 

y  —  c\e?  +  C<L(X*  +  3  x2  +  6x  +  6). 

o        _  x  i0         a* 

2. 

I  +  ^-2* 

+  150*-  183). 

3.  logjj/  =  c\ex  +  ^2^"*  +  xz  +  2. 

4.  j  =  fi  cot  *  +  ^2(1  —  x  cot  .r). 

• 

4. 

•^              *2 
y  =  ^i(^r2  —  i)  +  f&c. 

Section  62 

6. 

y  •=•  d*  (fi  +  c<i  log  #)  . 
^  =  x'*(c\  cos  *  +  c  2  sin  .*•). 

1.  y  —  c\  —  log  cos  (x  +  <:2). 
2.  y  =  (sin-1*)2  +  c\  sin"1*  +  <r2« 

7. 

jV  =  fix  +  ^C*"3  +  i). 

8. 

2J>  =  *(<!«*  +  <*-*)• 

a  —  y 

9. 

^  =  £!#"  cos  (a*  +  <:2). 

10. 

_y=  <TICOS  -+  ^  sin-- 

/                                                                                      9 

\                                        3 

Section  57 

-4*3log*J  +f2x 

3. 

r  -- 

y  —  x+ci  \  e    *dx  +  t%. 

j 

flX  +  f2 

4. 

y  =  (x  —  2)ex  +  fix  +  fZ. 

7.  y  =  c\  log  *  +  f%. 

x* 

5. 

y  =  I^*2  +  Wi  +  a2  +^2. 

12 

9-  y  =  log  cos  (<TI  —  x)  +  f2. 

Section  58 

10.  ^  =  —  +  <TlV.*2—  I  +  ^2. 

2 

2. 

(x-ay+y1=.b. 

11.  jj/  =  ^  +  fo  +  f3  log  *)  V^. 

3. 

y         4  ac2*?"5 

12.  ^  =  r  i  sin2  x  +  <r2  cos  x 
—  c%  sin2*  log  tan-. 

6        (i  —  0,?0*)2 

2 

4. 

«y  +  beax  +  2=0. 

13.  (rt)  The  circles   (*•  +  i:1)2+jj/2=c 

(^)  The  catenaries 

Section  60 

z+e2              *+ca 

2. 

xy  =  -  +  fi\ogx  +  f2. 

^=<?v  Cl  +  '  Cl  )* 

3. 

4 

(x  —  i~)2y  =  c  i  +  c-2X  —  cos  x. 

14.  (a}  The  cycloids 

4. 

xyVx2  —  i  =  ci^/x2  —  i 

x  +  <TI  =  fz  vers     •*•  —  •\/2fzy—y2, 

+  r2log(*  +  V*2-i)+f3. 

(3)  the  parabolas 

5. 

j^y2  =  ^i-*2  +  ^2-x1  +  ^3. 

(*  +  <Ti)2  =  2  f2>/  —  ^22. 

7. 

jy/  =  fjjr  —  c\  sin"1*  \/i  —  x2 

15.  The  central  conies 

8. 

y  —  c\  cos  .*2  +  c  2  sin  .r2  +  ^s*2. 

Cly2  —  £L  (x  +  f2)2=  i  ;  hyperbolas 

£ 

10. 

tan^  =  ^i  cot  x  +  <r2« 

when  £>o,  ellipses  when  ^<o. 

ANSWERS 


265 


17.  v={—=\/2g!  (cos6  -cos  a). 

dt 

18.  7;2  = 


(O, 


if  /fc  =  oo,  v  =  ^2gR,  which   is   7 
miles  per  second,  approximately. 

Section  64 


2. 


3.  .*  = 


x  —  czy 


9.  /  =  Clt, 
10.  j?-     =  fi (*- 


11.  x-y-t= 
12. 


Section  66 


Section  69 

The  path  is  a  parabola  lying  in  the 
vertical  plane  determined  by  the 
direction  of  the  initial  velocity.  Tak 
ing  the  initial  position  for  origin, 
the  horizontal  line  of  the  plane 
through  it  for  the  x  axis,  and  the  ver 
tical  line  through  it  for  the  y  axis, 
the  equations  of  the  path  are 
X  =  VQ(  cos  a,  y  = 


Or,  eliminating  /,  we  have 

Sx<i 

y  =  x  tan  a s z—  • 

'  2  z/o2  cos2  ot 


Or,  eliminating  /,  we  have 


an  ellipse  with  its  center  at  the  ori 
gin.  Since  x  and  y  are  periodic 

2  7T 

functions  of  period  —  ,  we  see  that 

the  motion  is  periodic,  the  period 
being  independent  of  the  dimen* 
sions  of  the  ellipse. 


266 


ANSWERS 


4.   2  kx  —  (ka  -f  v0  cos  a)  <?** 

•f  (ka  —  v0  cos  «) 
2/^  —  ^0  sin  «  (***  —  <?-**  ). 
Or,  eliminating  /,  we  have 


(x  sin  a  —  y  cos  a)2 ~ 

=  a2  sin2  a, 

a  hyperbola  with  its  center  at  the 
origin. 

q  =  a  cos 

r  =  c, 

where  a  =  v%2  -f 


Angular  velocity 

=  w  =  V/02  +  ^02  +  ^o2»  a  constant. 

Direction  cosines  of   the   instanta 

neous  axis  of  rotation  are  ^-,  S-,*^- 

u    (a    w 

6.  The  paths  are  the  curves  of  inter 
section  of  the  hyperbolic  cylinders 


The  time  is  given  by 


dz 


For  the  curve  through  the  origin 
c\  =  (%  =  o.  Hence  the  path  is  one 
of  the  lines 

a~±6~±c' 
The  time  in  covering  a  part  of  it  is 


-/  -±l(l       l\ 

°-^~U  *) 

=  ±l/i       i\±i(i       i\. 
«  VjJ'o     //       ^Uo     */ 


Section  74 


8.  v  = 


-  10 -f  $x-x2), 


5  A:'2 


10 


4.  v  = 


2.5      2.4.5.9 


2.4.6.5.9.13 


+ 


2.4 


2.3    2.4.3.7 

+...) 

; 


3.7." 
x* 


1.3.5.3.7.  ii 
5.  y  =  A  (6  -  4  x  +  *2) 
-f  £O 

Section  75 
,  i,  i,*), 


/          i         jr2  \ 
imit  F(a,p,---  -  ), 

«./3=°°      V          a       4ai8/ 


Limit 
Limit 


a,/3=<»  * 

Section  76 


2.  xq-yp  = 

3.  2  = 

4.  2  = 


6.  2  =  ^r/  +  jj/^  —  ^y^,  r  =  o,  /  =  O. 

7.  r  =  o,  j  =  o,  /  =  o. 


Section  77 


2. 


ANSWERS 


267 


4.  r-t=o. 

y 

Section  84 
L0fl_I,  iJflLa 

\x     y    y      z) 

i 

6.    X2  —  22  =  0  (*3  —  ^8). 
J 

7    2  —  nxev  4  —  cfie^v  4*  ^ 

2 

x-y          q  ' 
Section  79 

Section  82 

Section  83 

3.  ez  =  cxayl~a.     Since  the  equation  is 
linear  its  general  solution  is 

e-Xf\y}' 

8.  Vi  4-02  =  2V*4-tf.>'4-£. 

4.  2  =  cxay"^1  a*. 
5.  2  =  ax  4-  a  log  y  4-  c  .      Since    the 
equation  is  linear,  its  general  solu 
tion  is  2  =f(x  4-  logy)  . 
7.  az  —  I  =  cxeav.       » 
8.  4  02  =  (x+ay+  £)2. 
9.  2  =  aur  4-  «2J^  4-  *• 

9.  2  =  a  V^4-jv4-  Vi-tf2  Vx—y  4-  A 
10.  22  -  a2*2  =  (ay  4-  ^)2. 

2 

13.  (<z  —  i  ;  s  (.*•  4-  ay  4-  b)  —  a2  =  o. 

14.  2  =  aary  4~  &*  (x  4-  ^)  4-  & 
/  v\ 

11.  2  =  «:«•  +  by  4-  V02  4-^4-1.     Sin 
gular  solution  is  x2  +  y2  4-  22  =  i. 

12.  -  =  log  (x  +y  -  i)  4-  y  4-  b.     Since 

the  equation  is  linear,  its  general 
solution  is 
log  (x+y-i)+y=f(z). 

1O      „          ,.  f  r    |     ,.    i     fi\fny 

15.  *2  4-  *  V*2  4-  J2  4-  22  =  0  (  -  )• 
16.  2az=  (x  +  ay)2  4-  £. 
17.  (x  -  u)s(x  +y  +  z  +  u) 

j.0.  2  —  t  ^  -|-_y  -j-  a)*"*, 
15.  22  =  ^•20(^/)  4-     "V    >.  • 

*  Attention  should  be  called  to  the  fact 
that  no  unique  answer  can  be  given  for 
the  complete  solution.     Other  forms  just 
as  good  as  the  ones  here  given  may  be 
found  by  selecting  various  forms  for  the 
auxiliary  function  0. 

\  z  —  u     z  —  ul 
18.  The  family  of  planes 

2  =  ax  4-  V^2  —  I  —  <Py  4-  3  ; 
for  b,  a  fixed   constant,  the   corre 
sponding   planes    envelop    the   cir 
cular  cone 

whose  vertex  is  at  the  point  (o,  o,  b). 

268 


This  equation  is  a  particular  solu 
tion  for  all  values  of  b. 
19.  The  surfaces  of  revolution 


20.  The  family  of  planes 


ANSWERS 

5.  z  = 


where  the  fixed  points  are  (<:,  o,  o) 
and  (—  c,  o,  o),  and  the  constant 
product  is  &.  These  envelop  the 
ellipsoid  of  revolution 


Section  85 


4.  *  = 

5.  *  = 


3.  ,  = 


a.*:  +  by  +  cz) 
+  y(a 

Section  86 


Section  88 


Section  89 


2.  z= 


2.  2  =  01  O/  +  AT)    +  02  O  -  *) 

+  **•  (/-«). 

3.  2  =  0i  (j«r  +  j)  +  JC02  (x  +  y} 

+  03(^-^)   +^04(^-^). 

Section  91 

4.  a  =  00-*)  +^(2^+3*) 

l^  +  JL 


2  x)  + 


Section  92 


3.  z  =  0O  - 


4.  *  = 


-  sin 


Section  93 


Section  94 
1.  9  =  —\ogy  +  xy+  0O) 


30 


24 


4.  z  = 


7.  *=0(* 

8.  z  = 

9.  z  = 
10.  *=. 


+  (jr- 


INDEX 


The  numbers  refer  to  pages. 

The  following  abbreviations  are  used :  c.  c.  =  constant  coefficients,    d.  e.  =  differential 
equation.         1.  =  linear.         o.  =  ordinary,         p.  ^partial.         t.  =  total. 


Additive  constant,  2. 

D'Alembert,  233. 

Applications,  31-48,  55,  60,  67,  68,  74, 117- 

122,  146-148,  162,  163,  214,  229. 
Arbitrary  constant,  2. 
Auxiliary  equation,  92,  240,  246. 

Bernoulli,  20. 
Bernoulli's  equation,  20. 

Cauchy,  91,  92,  113,  164,  202. 
Cauchy's  linear  equation,  113,  178. 
Cayley,  75. 

Characteristic  equation,  92. 
Charpit,  218. 

method  of  Lagrange  and,  215. 
Clairaut,  25,  56. 
Clairaut's  equation,  56,  59. 

extended,  225. 

Commutative  operations,  97. 
Complementary  function,  91,  239. 
Complete  1.  d.  e.,  90. 
Complete  solution  of  o.  d.  e.,  5. 

of  p.  d.  e.  of  first  order,  213. 
Condition  for  exactness  of  o.  d.  e.  of  first 
order,  9. 

for  exactness  of  1.  o.  d.  e.,  138. 

for  integrability  of  t.  d.  e.,  77. 

for  relation  between  functions,  253. 

for  repeated  roots  of  an  algebraic  equa 
tion,  65. 

Consecutive  points,  70. 
Curve  of  pursuit,  147. 
Cuspidal  locus,  70. 


Darboux,  75,  202. 
Degree  of  d.  e.,  2. 
Derivation  of  o.  d.  e.,  3. 

of  p.  d.  e.,  196-201. 
Differential  equation,  i. 

of  a  family  of  curves,  31. 

of  simple  harmonic  motion,  118. 
Discriminant,  65. 

relation,  65. 

Envelope,  62,  63. 

Essential  arbitrary  constants,  3. 

Euler,  25,  91,  192. 

factor  or  multiplier,  25. 
Exact  differential,  8. 

d.  e.,  8,  ii,  137. 
Existence  theorem  for  o.  d.  e.,  164. 

for  p.  d.  e.,  203. 

First  integral,  52,  230. 
Functional  determinant,  253. 

Gauss,  192. 
Gauss's  equation,  193. 
General  integral,  6. 

General  plan  of  solution  of  o.  d.  e.  of  first 
order,  9. 

of  higher  order,  131. 
General  solution  of  o.  d.  e.,  5,  167. 

of  p.  d.  e.,  203. 

of  p.  d.  e.  of  first  order,  213. 
General  summary,  254. 
Geometrical  significance,  31,  61-63,  66.  69- 
71,  86,  157,  167,  203,  204,  213. 


269 


2/O 


INDEX 


Homogeneous,  function,  14. 

1.  o.  d.  e.,  89. 

1.  p.  d.  e.  of  first  order,  205. 

1.  p.  d.  e.  with  c.  c.,  239-245. 

o.  d.  e.  of  first  order,  14. 

t.  d.  e.,  81. 
Hypergeometric  series,  194. 

Integrable  t.  d.  e.,  76-84,  86. 

form  of  solution,  76,  85. 

method  of  solution,  80. 
Integral,  6. 

curve,  31. 

surface,  86. 
Integrating  factor,  8,  25,  141,  209. 

by  inspection,  23. 

of  o.  d.  e.  of  first  order,  indefinite  in 

number,  8. 
Integration  in  series,  164-195. 

of  o.  d.  e.  of  first  order,  169. 

of  o.  d.  e.  of  higher  order,  177. 
Intermediary  integral,  231. 

Jacobian,  253. 
Kowalewski,  202. 

Lagrange,  75,  103,  205,  213,  215,  218. 

method  of,  205. 

Lagrange  and  Charpit,  method  of,  215. 
Legendre,  114. 

Legendre's  linear  equation,  114. 
Leibnitz,  25. 
Linear  o.  d.  e.,  89. 

Cauchy's,  113,  178. 

complete,  90. 

general,  89. 

homogeneous,  89. 

Legendre's,  114. 

of  first  order,  18. 

of  second  order,  123-130. 

reducible  to  equations  with  c.  c.,  113, 
114,  126,  127. 

simultaneous,  with  c.  c.,  150. 

with  c.  c.,  91-122. 
Linear  p.  d.  e.,  general,  238. 

"  homogeneous,"  with  c.  c.,  239-245. 

non-homogeneous,  with  c.  c.,  246. 

of  first  order,  200,  205. 


Linear  p.  d.  e.,  of  second  order,  230. 

reducible  to  equations  with  c.  c.,  249 
Linearly  independent  functions,  90. 
Liouville,  142. 

Monge,  230. 
Monge's  equations,  231. 
method,  230. 

Nodal  locus,  69. 

Non-homogeneous  1.   p.  d.  e.  with  c.  c. 

246. 

Non-integrable  t.  d.  e.,  85,  86. 
Non-linear  p.  d.  e.  of  first  order,  211-226. 

Order  of  d.  e.,  2. 
Ordinary  d.  e.,  i. 

of  first  order  and  first  degree,  7-30. 

of  first  order  and  higher  degree,  49-75. 

of  higher  order,  89-148. 

reducible  to  1.  d.  e.  of  first  order,  20. 

system  of,  149-163. 
Orthogonal  trajectories,  38,  158. 

Partial  d.  e.,  i. 

of  first  order,  205-229. 

of  higher  order,  230-252. 
Particular  integral  of  o.  d.  e.,  6. 

of  1.  o.  d.  e.  in  general,  103-105,  125. 

of  1.  o.  d.  e.  with  c.  c.,  97-113. 

of  I.  p.  d.  e.  with  c.  c.,  243,  247. 
Particular  solution  of  o.  d.  e.,  5. 

of  p.  d.  e.  of  first  order,  213. 
Picard,  75,  164. 
Point  of  general  position,  32. 
Primitive  of  o.  d.  e.,  4. 

of  p.  d.  e.,  196,  199. 

Quadrature,  6. 

Reduction  of  d.  e.  to  equivalent  system, 

160. 

Regular  function,  203. 
Riccati,  173. 

Riccati's  equation,  28,  173. 
analogy  to  1.  d.  e.,  176. 
Roots  of  auxiliary  equation  repeated,  93, 

240. 
complex,  94,  242. 


INDEX 


2/1 


Separation  of  variables,  13,  25. 
Series,  integration  in,  164-195. 
Single-valued  function,  165. 
Singular  point  of  an  equation,  203. 
Singular  solution  of   o.   d.  e.,  63,  66-75, 
168. 

of  p.  d.  e.  of  first  order,  213. 
Solution  of  d.  e.,  2,  164,  202. 
Summary,  general,  254. 
Symbol  Z>,  89,  238. 

A  "4.  238. 

Symbolic  operator  (D —  a),  96. 
System  of  d.  e.,  149-163. 

general  method  of  solution,  149.  • 

of  1.  o.  d.  e.  with  c.  c.,  150. 


System  of  o.  d.  e.  of  first  order,  153. 
of  t.  d.  e.,  159. 

Tac-locus,  71. 
Tangent,  70. 
Total  d.  e.,  76-88. 

method  of  solution,  80-85. 

more  than  three  variables  involved,  83. 

simultaneous,  159. 
Trajectory,  38. 

Ultimate  points  of  intersection,  61. 
Undetermined  coefficients,  method  of,  107. 

Variation  of  parameters,  103, 


REFERENCES 


D'Alembert,  233. 

Boole,  101,  169,  174,  230. 

Cauchy,  113. 

Cayley,  75. 

Chrystal,  75. 

Craig,  91. 

Darboux,  75. 

Fine,  75. 

Forsyth,  174,  230,  244. 

Gauss,  195. 

Glaisher,  75. 

Goursat-Bourlet,  202,  203,  213. 


Hermite,  91. 

Hill,  72. 

Jackson,  122. 

Johnson,  174,  244. 

Lie,  25,  125. 

Liebmann,  75,  168,  169. 

Lobatto,  101. 

Marie,  233. 

Murray,  164,  244. 

Picard,  75,  164,  165,  169,  202. 

Schlesinger,  125,  141,  164,  183,  185,  194, 

Tait  and  Steele,  45. 


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